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Numerical methods for nonlinear integro-parabolic equations of fredholm type
AnI-Journal
computers 8 mathematics wlth~btiafm
PERGAMONComputers and Mathematics with Applications 41 (2001) 857-877
www.elsevier.nl/locate/camwa
Numerical Met hods for Nonlinear Integro-Parabolic Equations of F’redholm Type c. v.
PA0
Department of Mathematics, North Carolina State University Raleigh, NC 27695-8205, U.S.A. (Received March 8000; revised and accepted August 2000) Abstract-This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem ss well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm. @ 2001 Elsevier Science Ltd. All rights reserved.
Keywords-Integrc-parabolic equation, Finite-difference solution, Reaction-diffusion convection, Monotone iteration, Error estimate, Upper and lower solutions.
1. INTRODUCTION Integro-parabolic
differential
equations
of Fredholm
type arise from various fields of applied sci-
ences and have been investigated
by many researchers
therein).
Most
in the literature
equations
such as existence
dependent
of the discussions
solution
and uniqueness
(cf. [l-S]).
in the field (cf.
are devoted
of a solution
[l] and the references
to qualitative
and asymptotic
analysis
behavior
The work in [9,10] is related to finite-difference
of the
of the time-
solutions
of some
In this paper, we give a numerical treatment for a class of nonlinear integr+parabolic equations of F’redholm type under either linear or nonlinear boundary conditions. This treatment is based on the finite-difference method and the method of upper and lower solutions which lead to some monotone iterative schemes for the computation of numerical solutions as well as error estimates between the true finite-difference solution and the computed iterations. It also yields an existence-uniqueness theorem for the nonlinear finite-difference system.
special
type of integro-parabolic
equations.
0898-1221/01/t - see front matter @ 2001 Elsevier Science Ltd. All rights reserved. PII: SO8981221(00)00325-4
Typeset by -&S-W
c. v. PA0
858
The parabolic boundary-value problem under consideration is given in the form
ut-Duzr+V2Lz=f(Z,t,U,q*‘LL),
-a(“)u,(O, t) a(')u,(L,
+
t) +
(0 <
p(“)u(o,t) = g(O)(t, up,t)), pu(L,t) = gyt, u(L, t)),
<
L, 0 < t 5 T),
(1.1)
(0 < t I T),
4x7 0) = $4x), where D E o(z,t) positive constant,
2
(0 <
2
<
L),
and v E v(z,t) are the diffusion and convection coefficients, T is an arbitrary and g(O)(., u) and g(l)(., U) can be linear or nonlinear functions of u. The
function f(., U, w) is, in general, nonlinear in u and v with w = q * u, where
(q*u)(x,t)
G
IL
q(x,
x’, t)u(x’, t) dx’,
(0 Lx 5 L, 0
I: T).
0
(1.2)
It is assumed that the functions D(z, t), v(x,t), f(x, t,u,v), q(x,x’, t), g(O)(t, u), g(l)(t, u), and $J(z) are all continuous in their respective domains, D(s, t) is strictly positive in Z+ E [0, L] x [0, T], and q(z, z’, t) possesses the property
s L
and
q(x,x’, t) 2 0
q(x, x’, t) dx’ 2 1,
0
((x,t)
E DT).
(I-3)
The coefficients a@), pee) (e = 0,l) in the boundary condition are given either by ~(~1 = 0, pee) = 1 (Dirichlet condition) or by a (0 = 1 and pee) 1 0 (Neumann or Robin condition). In the case of Dirichlet boundary condition, the function gee)(t, u) is taken as g(l)(t) which is independent of u. To discretize problem (1.1) by the finite-difference method, we use the implicit scheme for parabolic equations and approximate (1.1) by a system of nonlinear finite-difference equations similar to that in [ll] for time-delayed parabolic equations. The use of the implicit scheme is critical in obtaining a meaningful finite-difference solution which preserves the qualitative properties of the original solution (see [12,13] and Remark 4.1). The purpose of this paper is to develop various monotone iterative schemes for the computation of the solution of the finitedifference system, including the existence and uniqueness of a finite-difference solution and error estimates of the iterative schemes. The iterative schemes are based on the method of upper and lower solutions which depend on the monotone property of f(., U, V) with respect to w. In these iterative schemes, the iterations are governed by a system of linear equations which can be easily solved by the standard Thomas algorithm. The plan of the paper is as follows. In Section 2, we formulate a finite-difference system of (1.1) by the implicit method and impose a basic hypothesis on the coefficient matrix so that the various types of boundary conditions can be included in the same framework of discussion. A monotone iterative scheme for the finite-difference system is given in Section 3, where the nonlinear function f (. , u, v) is considered nondecreasing in w. Similar monotone iterative schemes are given in Section 4 for some more general classes of nonlinear functions. In Section 5, we obtain an error estimate for each of the monotone iterative schemes. Applications are given to three model problems in the final section to demonstrate some techniques for the construction of upper and lower solutions and the implementation of the computational schemes.
2. THE Let h = Ax = L/M,
FINITE-DIFFERENCE
SYSTEM
Ic, = t, - tn_l, and xi = ih, and let A={(xi,t&
i=O,l,...,
M, n=l,...,
N},
Numerical Methods
859
where M and N are the respective number of divisions in [0, L] and [O,T].
When no con-
fusion arises, we write (i, n) instead of (zi,t*). At each mesh point (zi,tn), Di,, = D(%L)r Vi,n = V(Q, t,), and approximate the integral
we let ~i,~ =
U(Zi,k),
by the finite sum ((Cn) E A),
(2.1)
j=O
where u z (u~,~, . . . , UM+) T denotes a column vector in W”+’ weights possessing the property
and (pj}
is a set of quadrature
M O
51,
for (i, n) E A.
Pj 4i,j,n5 1,
c
(2.2)
j=O
The simplest choice of {pj} (q *u).
is pj = h for all j so that approximation (2.1) is a Riemann sum of
Define f(‘LLi,n,(4 * U)i,n) = f(G,L
U(G,L)r
gCO)(uo,n) = 9(o)(GI, u(O,L)),
(9 * U)(%,L)),
9(1)(Q&n) = 9%-6,
+,
L)).
(2.3)
By using the standard central difference operators
u4,nl =
(2)
&+J
( >
= +
(‘1Li+1,n - %-l,n), (2.4)
(%+1,n- 2ui,n + ui-l,n),
and the implicit method for parabolic equations, we approximate the integro-parabolic differential equation in (1.1) by the finite-difference equation IC,-%i,n -‘Iii ,n-l > -
a4,nl + ~zbi,?J= f(%,n, (q* U)i,n).
(2.5)
A simple computation shows that the above equation for the interior mesh points may be written as (1+2r,Di,n)ui,n =
U&n-1
-r, +
Di,,, + ++I,, K W(Ui,n, (!I * U)i,n>,
+ (Di,n- ++I+] (2.6) (i = 1,...,M-1),
where T,, = k,,/h 2. Similarly, by letting i = 0 and i = M in (2.5) and using the central difference approximation for ~~(0, tn) and u,(L, tn) in the boundary condition of (l.l), we obtain the following finite-difference approximations at the boundary points:
1 (q*4o.n)+r,@Do,n + h~o,n)dO)(uo,n>, = ‘1~0, n--l+ bJ(wLO,nt [
[
1 + 27-,Do,n + r,hP(“)(2Do,,z
1+ 2r,D~,,,
+ rJ@(‘)(2D~
+ hvo,n) uo,n - 2~nDo,nqn
,z - hVM,,J u~,n - 27-nJh4,nw--l,n
(2.7)
I
(see [12,14]). Th e initial condition is approximated by
w,o = %h,
(i = 0, 1, . . . , M),
(24
C. V. PAO
860
where ¢i = ¢ ( x i ) . System (2.6)-(2.8) gives a finite-difference a p p r o x i m a t i o n of the continuous p r o b l e m (1.1). If the b o u n d a r y condition at x = 0 is of the Dirichlet t y p e
u(O,t) = g(°)(t), then the first
equation in (2.7) is replaced by
(l+2rnDl,n)ul,n-r,~(Dl,n
h2'n)u2,n (2.9)
=Ul,n_l+knf(Ul,n,
(q* U)l,n)@rn (Dx,n + ~ ) g ( 0 ) ,
and is used for i = 1 in (2.6), where g(0) = g(O)(tn). Similarly, if the b o u n d a r y condition at x = L is given by u(L, t) = gO)(t), then the second equation in (2.7) is replaced by
(l+2rnDM_l,n)UM_,,n--rn(DM-l,n+hVM2--1'n)UM-2,
n (2.1o)
= U M - I , n - 1 + k n f ( U M - l , n , (q * U)M-1, n) + rn
DM-I,n
and is used for M - 1 in (2.6), where g(1) = g(1)(tn). In this situation, the finite-difference s y s t e m consists of M or M - 1 equations depending on whether the Dirichlet b o u n d a r y condition occurs in one or b o t h b o u n d a r y points. To include the various possible combination of b o u n d a r y conditions and some other possible forms of equations (such as equations in polar form) in the same framework of discussion, we consider a system of (p + 1) equations in the form (1 + rnai,n)Ui,n --
rn(b~,nUi-l,n + b~,nUi+l,n) = ui, n-1 + knf(ui,n, (q * u)i,n), (i = 1 , . . . , p -
1),
u (l+rnao,n)Uo,n-rnbto,nUl,n=Uo, n_l+knf(uo,n, (q*u)o,n)+rn bog (0) (o,n), (2.11) (1 + rnap,n)Up,n - rnbp,nUp-1, n = up, n-1 + knf(Up,n, (q * U)p,n) + rnb~g(1)(Up,n), ui,o=¢~, ( i = O , l , . . . p ) , where ai,n, hi,n, and b~,n are positive functions satisfying ai,n _> bi,n + b~,n and p stands for 1, or M - 2 depending on the type of b o u n d a r y conditions at x = 0 and x = L. Specifically, p = M if a (°) = a (1) = 1, p = M - 1 if a (°) = 1, a (1) = 0 (or a (°) --- 0, a (1) = 1), and p = M - 2 if a (°) = a(1) = 0. (In the latter case, the vector un defined below should be M,
M
-
un =
For each n = 1, 2 , . . . , define column vectors in R p+l by Un = (U0,n, Ul,n,... , Up,n) T,
Vn = (VO,n, V l , n , . . . , Vp,n) T,
(q * u)~ = ((q * u)0,n, (q * U ) l , ~ , . . . , (q * u)p,n) T,
(2.12)
F(un, Vn) = (ro(UO,n, v0,n), Fl(Ul,n, Vl,n),..., Fp(up,n, Vp,n)) T, where (q * u)i,n, i = 0, 1 , . . . ,p, are given by (2.1) (with M = p), and
Fi(Ui,n,Vi,n) = f(Ui,n,Vi,n),
(i = 1,... ,p -- 1),
Fo(uo,n, Vo,n) = f(uo,~, VO,n) + rnbog(°)(uo,,~), p,n) =
+
(2.13)
•
Define also a tridiagonal m a t r i x An by
ao,n -bl,n
b'
0
- - O~n
al,n "..
An =
• .,
"..
(2.14)
". ".
0
-bp,,~
'
--
b!p - - l ,
ap,n
Numerical Methods
861
Then we may write system (2.11) in the compact form
(I + rnAn)un = Un-1 + k n f ( u n , (q * u)n),
(n = 1, 2 , . . . ),
uo = ¢ ,
(2.15)
where ¢ = ( ¢ 0 , . . . , Cp)T. Notice from (2.1) that we may write (q* U)n = Qnun,
withQn=(pjqi,j,n),
i , j = O , 1,...,p.
(2.16)
It is clear from (1.3) and (2.2) that Qn is a nonnegative matrix and the sum of the elements of each row of Qn is bounded by one. To compute the solution of the finite-difference system (2.6)-(2.8), it suffices to do the same for system (2.15) with suitable conditions on An and F(un, vn). We impose the following basic hypothesis. HYPOTHESIS ( n ) . (i) The elements of An possess the property
bi,n>O,
b~,n>O,
and ai,n>_bi,n+b~,n,
((i,n) E h ) .
(2.17)
(ii) F ( u n , Vn) iS a Cl-function Of Un, Y n in a subset Sn O[R p+I. It is easy to see that by choosing the spatial increment h satisfying the relation h < 2 \lv~,nl] '
((i,n) c A),
(2.18)
condition (2.17) is satisfied by the coefficients in system (2.6),(2.7). In particular, condition (2.18) holds for every h > 0 if there is no convection (that is, v~,n -- 0). On the other hand, if the effect of convection dominates diffusion to the extent that it may require a prohibitively small h (that is, D~,n/v~,n is extremely small) an upwind differencing scheme leads to condition (2.17) without any restriction on h (cf. [12,15]). Hence, condition (2.17) can always be satisfied by system (2.6)-(2.8), including the Dirichlet boundary condition in (2.9) and (2.10). The hypothesis on F (un, vn) is also satisfied when f (., u, v), g(0) (., u), and g(1) (., u) are continuously differentiable in u and v. Condition (2.17) ensures that the matrix An is an M-matrix and for any rn > 0, the inverse matrix (I + mAn) -1 exists and is a positive matrix (cf. [16,17]). Moreover, the smallest eigenvalue #n of An is real and nonnegative, and if for each n, the strict inequality
ai,n > bi,n + b~,n holds for at least one i,
(2.19)
then #n > 0 (cf. [17]). The subset Sn in Hypothesis (ii) is taken as the sector between a pair of upper and lower solutions whose definition depends on the monotone property of F(un, Vn) with respect to vn (see (3.2) in Section 3). This property and the corresponding definition of upper and lower solutions are given in the following two sections. 3. M O N O T O N E ITERATION QUASI-MONOTONE NONDECREASING
FOR FUNCTIONS
To develop monotone iterative schemes for the finite-difference system (2.15), we require some quasi-monotone property of the function F(un, vn) with respect to Vn (but not with respect to Un). Recall that F(un, Vn) is said to be quasi-monotone nondecreasing (respectively, nonincreasing) in $n if ~v" > 0 (respectively, ~ < 0) for each i = 0, 1,... ,p and each (Un, vn) in $n, where Fi =- F~(ui,n,vi,n) is given by (2.13) and ~ v - ov,,,~" oF~ In this section, we present a monotone iterative scheme for quasi-monotone nondecreasing functions using either an upper solution or a lower solution as the initial iteration. The definition of upper and lower solutions for the present case is given by the following.
c. v. PA0
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DEFINITION 3.1. Let F(u,,v~)
be nondecreasing in v, for (un,vn)
E S,.
Then two vectors
iin, ti, in IW’+’ are called ordered upper and lower solutions of (2.15) if ii,, > ti,, and if (I+
r,&)f.&
2 %-i
+ W’(k, (n=
co > +,
(q * ti)n),
(3.1)
1,2,...),
and ti, satisfies the above inequalities in reversed order. The subset S, in the above definition is given by s, = { (un,vn)
E lRp+l x wp+l; ti, I un < tin, (q* ii)n I v, I (q* ti)n}.
(3.2)
For notational convenience, we set sJ&i’= {Un E lwp+i; ti, I lln I Gin}, s$
= {Vn E lwp+l; (q*ql
Iv,
Si,, = { (%n >%,n) E Iw x R
(3.3)
I (q*fi)n},
G,n I %,n I f-k,,, (4 * G)i,n I G,n 5 (4 * ii),,,},
where &,, and tii,n are the respective components of ii,, and ti,. function satisfying
Let Ti,n be any nonnegative
(3.4) and define matrices A, and I,, by A, = I + rnAn + k,J,,
rn = diag(yo,,, ~1,~~. . . , -yp,d
(3.5)
Then problem (2.15) may be written as Au,
= u,-1 + ICn[r,u, + F(u,,
(q * &)I,
(n=
1,2,...), (3.6)
us = $. Relation (3.4) implies that for every v, E SA2’, rnUn + F(u~, v,) L r,u:,
+ F(u;, v*),
when iii, > un 2 uk > i.&.
(3.7)
Moreover, by the nonnegative property of r, and Hypothesis (H), the inverse matrix
AZ1 exists and is a positive matrix (cf. [16,17]). Th’IS ensures that for any initial iteration un(0) , we can construct a sequence {uL~’ } from the linear iteration process A&“)
= u;_i
+ k,
[I’,ui”+‘)
up=q/J,(n = 1,2,. . . ),
+ F
(uim-‘),
(q * u);~-‘)
>I,
(3.8)
is the solution of (2.15) (or (3.6)) at the previous time step. It is wherem=1,2,...,anduA_i obvious that this sequence is well defined and can be easily computed by the Thomas algorithm because &
is a tridiagonal matrix (cf. [15,18]). Denote the sequence by {Eim)} if u$? = ti,,,
and by {ail”‘}
if u?’
= h,, and refer to them as maximal and minimal sequence, respectively.
We first show that if u;_i E St!, and is a solution of (2.15) at n - 1, then the maximal and minimal sequences possess the monotone property i& < r.&) < u(m+l) 5 iiim+i) 5 Eim) < - tin, - --n
m,n=
1,2 ,...,
(3.9)
and their respective limits lim aim) = 8,, m-WC
lim rain’ = By m-CO
(3.10)
exist and satisfy the equations (I+
r,&)u,
=
n;-1 + W’(u,,
uc = @.
(q * u),),
(n=1,2,...),
(3.11)
Numerical Methods
863
LEMMA 3.1. Let F(un, v,) be nondecreasing in v, for (un,v,J
E S,,, and let u;_~
E St?,
with UT, = 111. Then the sequences {iI&~‘}, {gim’ } governed by (3.8) possess the monotone property (3.9). Moreover, the limits iin and gn in (3.10) exist and satisfy (3.11). I& in addition, u; E S,$” and is a solution of (2.15), then gn I u; I i&.
PROOF. Letv&? = ii?)-ii:), wherei$ = iin. By (3.1), (3.5), and (3.8) (with m = l),
A,V(o) > tii,_l n = i&-l
-
u;_~+ k,
- u;_1,
(iin -ii;) >+
I’, [
F(ii,,
(q * a),) - F
(Tit’,(q * ix)~‘)]
(72 = 1,2,. . . ),
and w$@ = C-0- q!~2 0. Since by hypothesis iii,_1 2 u;_~, the positivity of A;ll implies that 2 0. This leads to ii:)
wi”)
gives &’ F(w,
v,)
2 L&?, where &’
2 EL’). A similar argument using the property of a lower solution = ii,.
Moreover, by (3.8),(3.7) and the nondecreasing property of
in v,, we have
which yields iii’) 2 gc’. This shows that --(O) u, > _ iii’) > &’ 2 &?. An induction argument, using property (3.7), leads to the monotone property (3.9). It follows from this property that the limits -& and gn in (3.10) exist and are in Si”. Letting m + 00 in (3.8) and using the equivalence relation between (3.6) and (2.15), show that tin and gn satisfy equation (3.11). To show the relation iin 2 ui when u: is a solution of (2.15), we observe that iiim’ = ut; = + and u: satisfies the equation dnu;: = u;-~
(n=
+ kn[I?nu; + F (u;, (q * ~*),a)],
1,2,...).
(3.12)
Substraction of the above equation from (3.8) yields
A,
(ii$$ -u*
n> =k,
[r, (~$‘+‘)-u$tF
(Bim--l), (q+E):m-l))-F(u;,
(q* u*),)]
, (3.13)
m = 1,2,. . . .
Consider the case m = 1. Since YiF) = tii, 2 u:, relation (3.7) and the nondecreasing property of F(u,, v,) in v, imply that &(iig) - uz) 2 0. This leads to ii;‘) 2 u:. Assume, by induction, that ail-‘) 2 uz. Then by (3.13),(3.7) and the nondecreasing property of F(u,,vn) in v,, we obtain &(Eim) - LIE) 2 0. This gives aim) 2 u;t. It follows from the induction principle that irim) 2 u; for every m. Letting m + 0;) and using (3.10) show that Q 2 u;. A similar argument gives ui 2 gn. This proves the lemma. I To show the relation iin = fin (= u;) and u: is the unique solution of (2.15), we set u n=max
{
iW,=max
$+,V&
(‘LLi,n,%,n)E %a,
i=O,l,...,
p
) >
2(%,&J
/ ; (W,n,‘Ui,n) E %I,
{I
(3.14)
i=O,l,...,p. 1
The following theorem gives the monotone convergence of the maximal and minimal sequences.
THEOREM 3.1.Let iin, tin be ordered upper and lower solutions of (2.15) where F(u,, v,) nondecreasing in v, for (u,, v,) E S, . Assume that Hypothesis (H) and the condition
Man + Mn) < 1,
(n=
1,2,...)
is
(3.15)
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PA0
are satisfied. Then a unique solution ui to (2.15) exists and is in S,(I). Moreover, the sequences {G
%${GP}
g’lven by (3.8) converge monotonically to u: and satisfy the relation
G, 5 &ml < u(m+l) 5 u:, < Eim+l) -< Eim) -< iin, - --n PROOF.
Assume, for the moment, that UC-~ E St?,
{aim)}, {ail’}
(m, n=
1,2 )... ).
(3.16)
with uz = $. By Lemma 3.1, the sequences
converge monotonically to their respective limits &
and u, that satisfy (3.11).
This implies that w, E iii, - u, satisfies the relation (I + r,A,)w,
= WY%
(q * %)
- F(gn,
(q * &)I.
(3.17)
Define the Jacobi matrices F,(u,,v,)
~Fo
z diag
-(
DUO,,
a&J
)I7 - (Up,nr 9.
UO,n,uo,n) 7..., (%,n7 1JP,?a aUP,n 8FP
'UP,73
dVP,n
By the mean-value theorem and (2.16), equation (3.17) may be written as (I + m&)wn
= k-&(&z,
~ln)w, + &(Cn, rln)(s * w)nl
= k&(&z,
rln) + F,(E,,
w-JQn]wv
.(3.18 (3.19)
where (&, vn) is an intermediate value in S,. Define P, = I+r,,A,
-kn[Fu.(Sn,rln)
+Fu(&z,rln)Qnl.
(3.20)
Then relation (3.19) is equivalent to P,w,
= 0,
(n=
1,2,...).
(3.21)
We show that under condition (3.15) the inverse matrix PC’ exists and is a positive matrix. Let P
= (pi”‘) *,n 14.7.
p, and let the diagonal elements of F,,(<,, 77,) be denoted = O,l,..., By (2.16) and the nonnegative property of F,,([,,qn), the elements of are given by d,,,(pj qa,j,d and are all nonnegative. In view of (3.20)
by do.n,k,n, . . . , dp,w the matrix 4,(L,vn)Qn and (2.14), the off-diagonal elements pi:) of P, possess the nonpositive property p$) < 0 when i # j. Moreover, by (3.14), p(n) x,2 > 1+?-,a. 2,n - k(o,
+ M,z . pi ~i.i,n),
j#i
j#i
It follows from Hypothesis (H) and conditions (2.2) and (3.15) that P
2 1 - Ic,(a,
+ M,)
> 0.
This implies that ply) > 0, ply) < 0 if j # i and P, is a strictly diagonally dominant matrix. It followed from these properties that P;’ exists and is a positive matrix (cf. [16,17]). Hence, w = 0 which shows that iin = gn. To complete the proof of the theorem, we show the existence of a unique solution by a ladder argument, starting from n = 1. Since UT,= + E Sil), Lemma 3.1 ensures that the sequences {-ii\“‘}, {Gil”‘} converge monotonically to their respective limits 31 and u, that satisfy (3.11) when n = 1. Lemma 3.1 and the above argument for n = 1 shows that El = ul and their common value is the unique solution of (2.15) at n = 1 and is denoted by UT. Knowing the solution u; and u; E Sill, the same reasoning shows that the sequences {iiim,“‘}, {u$~“‘} converge monotonically to a unique solution u; and u$ E Si’). to the conclusion of the theorem.
A continuation of the same process leads I
Numerical Methods
4. MORE
GENERAL
When the function F(u,,
v,)
865
QUASI-MONOTONE
FUNCTIONS
is quasi-monotone nonincreasing, the requirement of upper and
lower solutions is coupled and is given in the following.
DEFINITION 4.1. Let F(unrvn) be nonincreasing in v, for (un,vn) E S,. &, 6, are called coupled upper and lower solutions of (2.15) if G,, 2 G, and (I + T&J%
2 G-1
+ kJ(k,
(I + ~,x&)iL, I %,--I + kF(&,
Then two vectors
(q * cl),), (n=
(q * fi),),
1,2,...),
(4.1)
tie > + 1 tie. The subset S, in the above definition is given by (3.2) with respect to the above fin and ti,. Notice that the requirement of &, ti, in (4.1) is coupled, which leads to the following inter-related (but uncoupled) iteration process: A,&“)
= u;_~ + k, ~J%zcm-‘) + F @+l),
(q * fi)im-“)]
d,gkm)
= u;_~ + Ic, l?,&+‘) C
(q*E)im-‘)
E;m) = dm’ where u;_~
= $
m=
7
+ F
(
I&~-“,
1,2,...,
,
>I
,
(4.2)
(n=1,2,...),
is the unique solution of (2.15) in S:!,.
It is clear from Hypothesis (H) that
the sequences {YiLm)) , {I&~’ ) are well defined and can be computed by the Thomas algorithm whenever u;._~ is known. We show, as in Lemma 3.1, that these sequences possess the monotone property (3.9) and their limits _iin,gn in (3.10) satisfy the relation
(4.3) T&-J=L&
LEMMA 4.1. Let F(un,vn) with
UT, = +.
(n = 1,2,. . . ).
=+,
be nonincreasing in v,
Then the sequences
{Tiim)},
for (un,vn)
u in (3.10) satisfy relation property (3.9), and their limits ii,, _~ and is a solution of (2.15), then iin 2 u*71--T%’ > u PROOF. Let %Vi”)= iiF) -ii?)
and &’
= g?’
E S,,
and let u;_~
E St?,
{gimm,} governed by (4.2) possess
-&‘,
(4.3).
the monotone 16 in addition, I$ E S, (1)
where ii(‘) = tii, and gi”’ = ti,. By (3.5),
(4.1), and (4.2), we have -(O) w. = Co - $ 2 0, s (‘) = $ - ii0 >“o and d,%(0)n
2 (ti,_,
- u;_~)
+ k,
[
l?,
(
tin -iii’)
)
+ F (tin, (q * %)
- F @ho’, (q * u):‘)]
= tin-1 - u;_1 2 0, -&x,z to) _> (u;t_,
-i&-l)
+ k, [r,, (I& - fin) + F (,:‘,
= tin_1 - u;_1 2 0,
(q * ii(o))n)
- F (fin, (q * ti),,)]
(TX= 1,2,. . .).
By the positivity of Ail, i%$) 2 0, and vvi” 2 0 which yield ii:) 2 T&l) and &’ Similarly, by (4.2),(3.7) and the nonincreasing property of F(u,,v~) in v,, A, (Tit’ - &)) and@-&) &)
= k, [r” (@’
- I$‘)
+ F (Ti;‘,
(q * #)
- F (&“,
(q * ii):‘>]
1 &‘.
2 0,
= + - + = 0. This relation and the above conclusion lead to ?rF) 2 iii’) 2 > u(O) ,.... An induction argument shows that the sequences {i@)}, - _n foreveryn=0,1,2
c. v. PA0
866 {I&~)}
possess the monotone property (3.9). This implies that the limits -&,LI~ in (3.10) exist
and satisfy relation (4.3). Now if ut is a solution of (2.15) in Si”,
then UT, satisfies (3.12).
A subtraction of (3.12)
from (4.2) gives the relation A,,
($“‘-f)
=kn
sz, (upQQ) =I& +)
[rn(ti~m-l~-u;) ++in-‘),
(q * g)i@))-F(U;,
[rn(u;-lly,>
u*)&F(&-,
= Am’ = u;, =
+lqu;, m=1,2
$,
(q*
(..‘,
Consider the csse m = 1. Since EF) = tin 2 u: and II:’ the nonincreasing property of F(unrvn) in v, that A, (Ii?) - u;) This leads to Zi,‘) 2 uz 2 r&l’.
2 0
and
(n=1,2
(q *
U*)n)] ,
(q*li)g-‘))I , (4.4) )... ).
- i& I uz, we see from (4.4),(3.7) and
A, (u; - II:‘>
2 0.
An induction argument using relations (4.4),(3.7),
nonincreasing property of F(u,, v,) in vn, yields i@) 2 ui m -+ co shows that & 2 ui 2 LIP. This proves the lemma.
(m) for every m. 2 II~
and the Letting I
Using the functions cr, and M, given by (3.14) with respect to the present coupled upper and lower solutions, we have the following analogous convergence theorem as that in Theorem 3.1. THEOREM 4.1. Let iii,, tin be coupled upper and lower solutions of (2.15) where F(un, v,) is . . nonincreasmg In v,, and let Hypothesis (H) and condition (3.15) be satisfied. Then a unique solution ult, to (2.15) exists and is in Si”. Moreover, the sequences by (4.2) converge monotonically to u; and satisfy relation (3.16).
{ii~m)},{~k~,m)} governed
{gi”‘} converge PROOF. In view of UC;= @ E Si” , Lemma 4.1 implies that the sequences {Ei”‘), monotonically to their respective limits El and II, that satisfy (4.3) at n = 1. Let WI = Ii1 - II,. By (4.3) and the mean-value theorem,
(I + ~lAl)wl = h[J’(%, (q * 41 - WI, (q *WI = h[&(&,m) + (--~d~~,m)Ql)lw. Since the above relation is in the same form as that in (3.19) (with n = 1) where F,(&,ql) is replaced by (-F,(
The I
The method for quasi-monotone nondecreasing and quasi-monotone nonincreasing functions can be extended to a more general class of functions in the form
qu,,
v,) =
F(‘)(u,, v,) + F(2+hvn),
(4.5)
where F(l)( II,, v,) is nondecreasing in v, and Fc2)( u,, v,) is nonincreasing in v, for (u,, v,) E s,. The subset S,, is again given by the sector between a pair of coupled upper and lower solutions which are defined as follows.
NumericalMethods DEFINITION 4.2.
867
are called coupled
be given by (4.5). Then two vectors i&,ii,
Let F(un,vn)
upper and lower solutions of (2.15) if ii,, > iin and
(I + rnA,)tin
2 tin-1 + k,, [F(l) (ii,, (q * a),) + F(‘) (fin, (q * ti)n)] ,
(I + r,&)ti,
5 L-1
(4.6)
+ k, [F(l) (iin, (q * ti)n) + Ft2) (tin, (q * ti)n)] , (n=
Go r + r Go,
1,2,...).
We again define the sectors S,, SLe’, and Si,,, e = 1,2, by (3.2) and (3.3) with respect to the above pair of upper and lower solutions. For each e = 1,2, define yz’z, oie), and Mie’ by (3.4) and (3.14) where Fi(ui,+, vi,,,) is replaced by F!e)(~i,n, Q+). z
The functions y,!,:, r,!“,’ are chosen
such that ~j,z + yj,z 2 0. Define also A, = I + T,A, + k, (I’~‘) + I$))
I’$
,
= diag (-$A,.
. . ,$A)
(e = 1,2).
,
(4.7)
Then by Hypothesis (H), the inverse A,1 exists and is a positive matrix (cf. [17]). Moreover, for every v, E SL2’, I’f)u,
+ FCe)(u,, vn) 1 rr)uk
Using ~2) = tin and &” {irim)}, {&?}
+ FCe) (uh, v,) ,
when tin 2 un 2 uk L tin,
(a = 1,2).
(4.8)
= t, as a pair of coupled initial iterations, we construct two sequences
from the linear iteration process
J&m)
I:) + IF) ~im-1) + ~(1) i$+i), > ( K +1”(Z) Ehm-11, (q * u)C$-r’ (
= u;_l
(q * i-j)im-i)
+ Ic,
)I
A,Llim) =u;_1 +k,
[ (rg) + ri2))&m-l)+ F(l) (&+l),
gg4, (q* qp-1) ( )I $4 = @’ = $ 1 m= 1,2,...,
>
(q* g)$yl’>
(4.9)
+Ja
(n=
1,2,...),
where u; = $J. It is clear that these two sequences are well defined and can be computed by the Thomas algorithm. In the following lemma, we show the monotone property of these sequences. LEMMA 4.2.
be given by (4.5), and let u;_~ E St?,
Let F(u,,v,)
with ug = $J. Then the
sequences {ii~m)},{~&m”’} governed by (4.9) possess the monotone property (3.9). PROOF Let F(O) = iji”) - ii(l) do) = uil) - g:‘,
where ii(O) Tl = fi VIand
and I’* 71= j$,‘) + rr’,
I&” = a 7L*By (n4.6), (4.7), annd;47-d;, we have &w(O)n > - fin_1 - u;_~ + k,
[
r:, ( tin- iiF))
+ FC2) (ii 7LI(q * G),) - FC2) (fi?‘,
+ F(l) (ii,,
(q * a),)
- F(l) (ii?‘,
(q * i?)?‘)
(q * E&c’)]
= tin-i - I$&_, 2 0, d,vvi’)
> - u;_i
- i&-r+
+ Fc2) (@, = u;_l
-ti,_i
The positivity of A,1
k,
[
r:, ( lly -
(q * n)?)) 10,
^ un) + F(l) ( a;),
- FC2) (L (n=
implies that Vt)
(q * s)p’)
- F(l) (tin, (q * ti)n)
(q * a),)]
1,2,...).
2 0 and v&c) 1 0 for n = 1,2,. . . . It is obvious that
W(O) - 0 and w-, = $ - tia 1 0. This shows that i$) 7z = tic - Q >
5 @‘I
and &’
5 ~2)
for
868
c. v. PA0
n = 0, 1,2, . . . . Moreover, by (4.8),(4.9) have ZF’ = &)
and the monotone property of Fte)(un, v,)
in v,,
we
= $J and
A, (i%(l) -Q)
= k, p;
(@
+Fc2) (Ii;), This leads to ii:)
2 gc’
-g?‘)
+ F(i) (ii?‘,
(q * I#)
(a * Q’)
- F(l) (ll?,
- Fc2) (I$“, (qci):))]
(q * t&c’)
2 0.
for n = 0, 1,2, . . . . The monotone property (3.9) follows by an induction
argument.
I
Because of the monotone property (3.9), the limits iin and gn in (3.10) exist and satisfy the relation ti, 15 gn I iin I tin. Letting m + co in (4.9) and using relation (4.7) show that E~ and an satisfy the equations (I + r,A,)En
= u:-~ + k, [F(‘)(G,
(q * a),) + F’2’(Ti;2, (q * a)n)] ,
(I + ~,A,)~~
= u;_*L-_I + k, [J’%n,
(q * u)n) + F(2)(~nr
Es = L& = ?j$
(n=
In the following theorem, we show that En = u,(=
(q * i&z,]
(4.10)
,
1,2,...).
uz) and u: is the unique solution of (2.15).
THEOREM 4.2. Let F(u,,v,J be given by (4.5), and let iii,, ii,, be coupled upper and lower solutions of (2.15). Assume that Hypothesis (H) and the condition k, (o$” + ,i2) + Mi1) + A4i2)) < 1,
(n = 1,2,. . . )
are satisfied, where CL” and MAe’, .C = 1,2, are given by (3.14) with respect Then a unique solution {xim’}
II; to (2.15) exists and is in SA”.
(4.11) to Fce)(un,v,).
Moreover, the sequences {iiim)},
governed by (4.9) converge monotonically to UC and satisfy
(3.16).
PROOF. Consider the case n = 1. Since UT,= $ E S,$‘), Lemma 4.2 ensures that the sequences {a’;“‘},
{ui”‘}
possess the monotone property (3.9), and therefore, the limits ili,u,
exist and satisfy (4.10) at n = 1. Let w$l) = iii’) -gi’). mean-value theorem.
(I+ n&b,
(l) = ICI [F(‘)(i&, (q*ii)l) - F(%,, +F(2)(&, (q* u)d - Fc2h,
in (3.10)
Then by (4.10) (with n = 1) and the
(q * 41) (q * n),)]
where (Cl, ~1) and (G, 77:) are some intermediate values in Si. Since the above relation is the same as that in (3.19) (with n = 1) when F, and F, are replaced, respectively, by Fil’ + Fi2’ and F$‘) - Fi2), and since Fil’(&,
T.II)+ Fi2’ (& 77’) I ai’) + a?),
IF;‘)(Em)
- FJ2’ (S;,s;)l
5 M,(l) + M,(2),
we conclude from the argument in the proof of Theorem 3.1 and condition (4.11) that iii = a,(= UT) and UT is the unique solution of (2.15) at n = 1. Moreover, relation (3.16) holds at n = 1. Knowing the solution u; and UT E Sil) the same ladder argument as that in the proof of Theorem 3.1, using the conclusions in Lemma 4.2, shows that a.unique solution ui E SA” to (2.15) exists and the sequences {YiAm)}, {@“‘} relation (3.16). This proves the theorem.
converge monotonically
to UC and satisfy I
NumericalMethods
869
REMARK 4.1. (a) When f(z, u, V) = f(z, U) is independent of v, the definitions of ordered and coupled upper and lower solutions coincide, and problem (2.15) is reduced to a finite-difference system for standard parabolic boundary-value problems. Hence, the results in Theorems 3.1, 4.1, and 4.2 are directly applicable to this class of parabolic equations (cf. [13,19]). (b) In the formulation of the finite-difference system (2.6)-(2.8), we have used the implicit method for parabolic equations.
This approach ensures that the computation
of the
iterations aim) and girn’ from any one of the linear iterative schemes (3.8), (4.2), and (4.9) is unconditionally stable with respect to the mesh sizes k, and h (cf. [15,18]). This fact and the consideration of the variable time-increment k, make it more attractive in the computation of numerical solutions with large time t,, especially in relation to the asymptotic behavior of the solution. It should be pointed out that if the explicit or a semi-implicit method is used, then the resulting numerical solution may be incorrect, misleading, or incompatible with the solution of the continuous problem. Some examples for this situation can be found in [13,14].
5. ERROR
ESTIMATES
AND
NUMERICAL
STABILITY
It is seen from the monotone convergence of the maximal and minimal sequences {x;~)}, {L&~)} that given any EO> 0 and any norm in lfUP+l,there exists an integer m* = m*(co) such that lliml - u:, < when m 1 mm, - iiim) -l&rim’ < EO, I/ II II I/ where uimn,stands for either ai*) u:
and the mth approximation
or zLm’. This shows that the error between the true solution L&“’ is bounded by EOwhen m 2 m*.
computation, the true solution ui_i
However, in actual
in the iteration process is taken as ui:‘i
for some m’. This leads to an error between the theoretical and the computed mth approximation for each m and each n. In addition, there may be round-off error and possibly initial error in the computation of L&~). The aim of this section is to give an error estimate between the true solution uz and the computed mth approximation vvimm,for each of the iterative schemes (3.8), (4.2), and (4.9). Since for any norm in Rpfl ,
the convergence of {@“‘}
to ui implies that
wp - u:,II<- IIwp - up) II+
E,,
when m 2 m* .
(5.1)
Hence, to investigate the error between the computed mth approximation whrn”’ and the true solution ufi, it suffices to obtain an error estimate between wnCm)and uLmm, for a suitably large m 2 m* . We do this for each of the three types of quasi-monotone functions F(u,, v,) treated in the previous two sections. For technical reasons, we assume that & = A and k, E k are independent of n, and choose ^fi+ = 7 so that I’, E l? = yl and A, = A. 5.1. Quasi-Monotone
Nondecreasing
Functions
For quasi-monotone
nondecressing functions, the iteration process is given by (3.8) and the computed mth iteration w,,(m) satisfies the equation dwim) = u;__~ + k I’w(“+~) +F 71
w(“+ It
(q * w)(“‘--~) n
>I
+ eim),
(5.2)
c. v.
870
PA0
where WV;‘) is either iin or ti, and eirn’ denotes the combined error due to round-off and the approximation of uz_. 1 by w;~‘). Let e, and e, be the maximal round-off error and the maximal allowable error between the theoretical and computed mth approximation, respectively. Then by the convergence of the maximal and minimal sequences in Theorem 3.1, there exists an integer m* such that ei? I/
II
when m 2 m*,
(n=
1,2 I...) N).
(5.3)
Let [lQnll denote the operator norm of the matrix Q,, and define (5.4) where y, u,, and ikf, are given by (3.4) (with y > ~i,~) and (3.14), respectively, and p > 0 is the smallest eigenvalue of A. In terms of the parameter wn, we have the following error estimate between the true solution and the computed approximations. THEOREM 5.1.
Let II; be the true solution of (2.15) where F(unr vn) is nondecreasing in v,, and
n = ti,. Assume let {w;~“‘} be th e computed sequences from (3.8) with either w,,(‘) = ti n or w(O) that Hypothesis (H) and condition (3.15) are satisfied. Then there exist an integer m* and a norm in KY’+’ such that
II,pm, - II e,, u;
< - 1-
kw,
for all m 2 m*,
(72 = 1,2,. . . ) Iv),
(5.5)
provided that kw, < 1. PROOF. In view of (5.1),
it suffices to obtain
an error estimate
for I[w~~,“’- uL~) 11 with
m > m* and a sufficiently small E,, where ui”’ = who’. Let Ehm’ = wimLm, - uimm,and IELm’I , where E!“) 2,71) i=O,l,..., = (pm)) p, are the components of Ekm’. By subtract0,n ) ... 7IJ$?I)T 1 ing (3.8) from (5.2) and using the mean-value theorem, we obtain
w;“+, - F
(uim-‘),
=k [I?+ F,(&,vn)
(q *
u)$-
+ F,(&x,
(q * w)$‘+l)
>I+ eim)
>
wdQn1J$‘+l) + ek’?
where (&, q,) is an intermediate value. In view of (3.14), y + ~7~2 0, and the positivity of d-l and Qn, the above relation yields the estimate
lEim)) I A-’ {k [(y + a,)1 + I&Q,]
IEhm-l)l + ieim’i},
(5.6)
and etm), i = 0 1 .., p t are the components of eim,“‘. Since where je?)I = (Je(m)I 0,n i ...) leimn)I)T I the spectral radius of A-’ is bounded $/3-l, whek /3 is given by (5.4), we see from (5.3) and (5.4) that given any small 6 > 0, there exists a norm in IF’+’ such that (5.7) for all m 2 m*, where IIQnII is the operator norm of Qn (cf. [20]). Let b, = kw,, E = e, + e,, and choose E sufficiently small, where w, is given by (5.4). Since ,8-l 5 1, the above relation yields l/E~~)II~b,I/~~~-‘)lj+e, In view of EL’) = wi”’ - I$’
whenm>m*,
(n=1,2,...,~).
(5.8)
= 0, an induction argument in m (with n fixed) gives
IJE$-~)I/< (~~~-1)+b~~-2)+...+1)~<(1-b,)-1~,
(m.
The above inequality and relation (5.1) (with EOsufficiently small) leads to estimate (5.5). proves the theorem.
This I
NumericalMethods
5.2. Q&G-Monotone
Nor&creasing
For quasi-monotone
871
Functions
nonincreasing functions, the computed mth iterations &n,“) and P&~’
satisfy the equations A%$$,“) = u;_i
+ k l?i+@)
[
+ F Wim-l), (q * w);y))] (
+ i$‘, (5.9)
w(“+l), (q * .iiiT)im-l)_ + girn), AELm) = u;_~+ k 13vim-‘) + F (_?z [ >I
where CZirn)and TV”’ denote th e combined round-off and allowable errors between aim) and i~$~) and between r&’ and y&?, respectively. By the monotone convergence result of Theorem 4.1, there exists an integer m* such that Il~~~)ll+Il~~m)ll
forallm2m*,
(5.10)
(n=1,2,...,N).
In the following theorem, we give an analogous estimate as that in Theorem 5.1. THEOREM 5.2.
Let UC be the solution of (2.15) where F(h,vn)
is nonincreasing in v,,
and = tii, and yip’ = ti,.
let {iVim)} and {ail’}
be the computed sequences from (4.2) with w?) Assume that Hypothesis (H) and condition (3.15) are satisfied. Then there exist an integer m* and a norm in lRP+l such that
IImy-,;II+II&Lu;II
(n=1,2
forallm>m*,
l-kw,
,...,
N),
(5.11)
provided that lcw, < 1. PROOF. Let ELm’ = dnm) - grn) and B,“’ = ail’ applying the mean-value theorem, we obtain
d$?
= k ki$m-l)
+ F,(&,qn)$~-‘)
A@’
= k rEi+
+ F, (<;, y/;)&+‘)
where (L,s)
[
and (C,,$,) 1E;“‘l
Id-‘{
Let Eim) = Ekm’ +&ml
-ghm’.
By subtracting (4.2) from (5.9) and
+ F&,
,n)Q,&m-l)]
+ F,, (t;, q;) Q,&&‘]
+ z$,~‘, + e,“‘,
are some intermediate values. The above relation implies that
(~~~-l'l+M,Q,I~~~-l)I] +I+'I},
k[(r+cm)
and IeLm’1= Ieim)I + I@’
I. Then addition of the above equations leads
to
IIE~~)/ I A-'{ [k(y+c,)I+
ACLQ,](~;y+~)l+ leim)l}.
(5.12)
Since the above inequality is of the same form as that in (5.6), we conclude from (5.10) and the argument in the proof of Theorem 5.1 that (5.11) holds. I 5.3. The Extended If the function F(h, the relation
Quasi-Monotone
Functions
v,) is given by (4.5), the computed mth approximations ~v:~),v&~) satisfy + k r’+$+‘) + j’(i) +m-‘), [ ( (q * w)(m-i) +F@) ++‘), T& ( ?a
&+-km’ = u;_l
&IL*’ =u;__~ +k l?*,(m-l) [
+Ft2)
(
&‘+l),
>I+i+’
+ F(l)
(q * W)im-”
(m,n=l,2
(q*~)(m-‘) n
(
E;~-“,
11 ,... ),
+ &“’
(q &$.,m-l)
>
>
(5.13)
c. v. PA0
872 where
I’*
=
l?(l)
+
and BLm)and c&~’ are the combined errors satisfying (5.10). Define
l-‘c2)
u; = I$) + aA2), p* = 1+ qp
M,* =
MC) + M(2) 7% 7 w; =
+ y*),
y*
=
p
+
p,
where crie) and iMAe’, C = 1,2, are the functions appearing in (4.11) and y([) 2 7::. have the following error estimate. 5.3. Let I$
THEOREM
be the solution
(5.14)
(y*+ 6%;*WQnll),
of (2.15) where F(un,vn)
Then we
is given by (4.5), and let
} be the computed sequences from (4.9) with W$,? = iin and VV,? = ii,. Assume {~~~‘},{Fv~? that Hypothesis (H) and condition (4.11) axe satisfied. Then there exist an integer m* and a norm in IV+’ such that
I/ II&Lu:,
I/
$$L,;
provided
+
II<-+ -
1 - lcw;
for all m 2 m*,
(n = 1,2,. . . , N),
(5.15)
that kwz < 1.
PROOF. Let E’“’
= Ktrn) -Ecrn) and Ecrn) = yim’ -I_&~). By subtracting the equations in (4.9) from the corresiondingnequatikrs in (cy3) and applying the mean-value theorem, we obtain
where (&, Tjn), etc. are some intermediate values. In view of the nonnegative property of A-‘,
(I’*
+ F$l’ + Fi2)), and Qn, the same reasoning as that in the proofs of Theorems 5.1 and 5.2 shows that
Addition of the above inequalities leads to the relation
IEkm)) 5 A-’ {k[(-I* + a;) I + M;Qn]
IE;m-l)l
+
Ie~m’l} ,
(5.16)
where Eim) = ELm’ + E,“), eP’ = Z&m’ + c&~) and y*, a:, and M,* are given by (5.14). Since (5.16) is of the same form as that in (5.6), we conclude from the definition of w;*Land the argument in the proof of Theorem 5.1 that relation (5.15) holds. I The error estimates given in Theorems 5.1-5.3 are based on “internal” perturbations (round-off error and the approximation of u:__~ by w,(m’)) and the initial perturbation eim’. These estimates yield the following numerical stability of the iterative schemes. THEOREM monotone
5.4. Let the conditions in Theorems 5.1-5.3 be satisfied for the corresponding function F(un, v,). Assume that there exists a constant 6 > 0 such that knw,
5 1- 6
and
knwz I 1 - S,
for all n = 1,2, . . . ,
quasi-
(5.17)
NumericalMethods
873
respectively. Then the corresponding iterative schemes (3.8), (4.2), and (4.9) are numerically stable with respect to combined internal and initial perturbations. PROOF. Under condition (5.17), estimates (5.5), (5.11), and (5.15) in Theorems 5.1-5.3, respec-
tively, imply that the error (Iji$“) - uzll + I&*’ -
It is easy to see from (5.4) that the requirement kw, < 1 in Theorems 5.1 and 5.2
is equivalent to k(o;, +
MdlQ,ll - P>I 1.
Since /_L2 0, the above relation follows from condition (3.15) if IIQnII5 1. Similarly, the requirement kwg < 1 in Theorem 5.3 follows from condition (4.11) if IIQnll < 1.
6. APPLICATIONS It is seen from Theorems 3.1, 4.1, and 4.2 that the main conditions for the monotone convergence of the maximal and minimal sequences {%@)}, {&m’} are the quasi-monotone property of the nonlinear function F(h,vn) and the existence of a pair of the corresponding upper and lower solutions. In the computation of these sequences, it is also necessary to choose l?, and k, so that condition (3.15) or condition (4.11) are satisfied. In this section, we consider three model problems (with linear boundary conditions) and give some basic techniques for the construction of upper and lower solutions and the choice of I’, and k,. These model problems demonstrate each of the three types of reaction functions treated in Sections 3 and 4. PROBLEM (a). The first model problem involves a Michael-Menton type of nonlinear function arising from enzyme-substrate reactions. This problem is given in the form (cf. [1,21])
ut - Du,, = au +
u(0, t) -
a(O)u,(O, t) =
‘4q* 4 1+q*u
g(O)(t)
‘ll(G0)
S(Ga
+
u(L, t) +
1ti(z),
(0 <
cA,(L, (0 <
2 <
L,
t) = g(l)(t)
t >
9
O),
(t > 01,
(6.1)
L),
2 <
where a zz a(t) is an arbitrary continuous function of t, and b E b(t) and ~(2, t) are continuous nonnegative functions of t and (z, t), respectively. For physical reasons, we assume that g(O)(t), g(l)(t), and $(cc) are continuous nonnegative functions in their respective domains. By the finite-difference approximations obtained in Section 2, a finitedifference system of (6.1) is given in the compact form (2.15) where the coefficient matrix A, possesses the property in Hypothesis (H)-(i). The function F(un,vn) is given by (2.12) with
Fi(ui,,, 2liJ = anu+ + *
z,n
+ Si,n + B&n,
(i=O,l,.*.,
p),
(6.2)
where & ii,*
= 0,
ifi=l,...,
=
p-l,
if i = p, and (n=
(6.3)
1,2,...).
Clearly, F(u,,v~) is a @-function for all u, 1 0, v n 1 0, and is nondecreasing in v,. This implies that the requirements of upper and lower solutions are given by Definition 3.1. It is obvious from the nonnegative property of ~i,~, _1cli,and gc’ (! = 0,l) that Q, = 0 is a lower solution. Let b, = (bn,. . . , b,JT,
ST3= (Sop, *. * 7Sp,n )+T
&I = (iO,n, 0,. . . , 0, &l)T,
c. v.
a74
and let U,
be the positive
solution
PA0
of the linear problem
(I + T-,&&J, = C-1 + kn[anUn + (bn + sn + in)], By Hypothesis
(H)-(i),
the existence
of a positive
solution
u,=ylJ.
to (6.4) is ensured
(6.4)
if k, a, < 1. It is
easy to verify from wi+/( 1+ vi,,) I 1 for wi+ _> 0 that fii, = U, is an upper solution. that the pair tin = U, and ii,, = 0 are ordered upper and lower solutions.
This shows
Since by (6.2),
the functions in (3.4) and (3.14) may be taken as yi+ = 0, CJ, = a,, that rn is the zero matrix and condition (3.15) becomes
Man + W < 1,
and M, = b,. This implies
(n = 1,2,...).
(6.5)
As a consequence of Theorem 3.1, we have the following conclusion: under condition (6.5) the model problem (6.1) has a unique finite-difference solution u; 2 0. Moreover, the sequences from (3.8) with -iI) = U,, 3:’ = 0, and I?, = 0 converge monotonrelation (3.16) with 11, = U, and ti, = 0, where U, is the positive
{@)}, {I_&~“‘}obtained ically to E: and satisfy solution
of (6.4).
PROBLEM (b) . As a second example, the form (cf. (1,6]) ut - Duzr =
we consider
a modified
u(a - bq * u) + ~(5, t),
diffusive
(0 <
5
logistic
equation
given in
L, t > O),
<
(6.6)
where the functions a, b, and s and the boundary and initial conditions are the same as those in (6.1). In this model, the function F(un,vn) is nonincreasing in v, for all (u,,v~) 2 (O,O), and its components are given by
Fi(ui,n, WUi,n) = Ui,n(G
-
(i = O,l,. . .,p).
bn~,n) + si,n + h,n,
(6.7)
In view of Definition 4.1 and Fi(O, v,) = .si,* + &,,, > 0 for any vn, the pair iin, ii, with ti, = 0 are upper and lower solutions if fin 2 0 and
(I + r,A,)iii,
2 tin-1 + k,F(k,
By (6.7), it suffices to choose tin = U,, lem (6.4) with b, = 0. Moreover, from
where U,
z(Ui,n,wi.n) =an - bnwi,n, the functions
0),
is the positive
z
(Ui,nr
solution
of the linear
prob-
wi,n)= -bnui,n,
in (3.4) and (3.14) are given by TOI
=x
=
max{O,
b, (q*q,
-a,},
where u’, is the maximum value of the components and A4,, condition (3.15) is reduced to
k, (an+ bnrn) < 1,
it& = b,u,,
an = a,, Vi,, of U,.
With
(n = 1,2,. . . ).
(6.8)
the above choice of on
(6.9)
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875
By an application of Theorem 4.1, we have the following conclusion:
under condition
(6.9),
equation (6.6) (with the boundary and initial conditions in (6.1)) has a unique finite-difference solution ui 2 0. Moreover, the sequences {@‘)},
{ail”‘}
obtained from (4.2) with $1
tin(‘) = 0, and I+, = ~~1 converge monotonically to ut and satisfy relation (3.16) the positive solution of (6.4) with b = 0. PROBLEM (c).
= U,,
where U,
is
When the modified reaction function in the diffusive logistic equation is replaced
by (g * u) (o - bzl), equation (6.6) becomes Ut
-
Du,,
(0 < x < L, t > O),
= (g * u) (a - bu) + s(x, t),
(6.10)
where a, b, and g are the same as that in (6.1). In this case, the components of F(un,vn)
are
given by (6.11)
Fi(ui,n, vi+) = u~i,n(on- b, ~i,n) + si,n + Gi,n.
Although the above functions are nondecreasing in ui,,, if ?.+ I an/b,,, this latter requirement imposes a very restrictive condition for the construction of a positive upper solution. To overcome this, we define
F!‘)(u. r,nr *
v.*,n
) =u n 21.r,n
and write F(un, v,) = F(‘)( u,,v,)
+sj,+ij.
*,n
7
+ F(‘)(u,,v~),
1
$d2’(u. r,n, z
v.r,n ) = -b n u. *,n v:l,lz
where F@(h,
v,),
(6.12)
e = 1,2, are the vectors
,r,n, vi+). Clearly, F(‘)(u,,v,) is nondecreasing in v, and Ft2)(u,, v,) with components F!“(u. * 2 (0,O). It is easy to verify from Definition 4.2 and (6.12) is nonincreasing in vn for (h,v,J that the pair tin = U, and ti, = 0 are coupled upper and lower solutions if U, is nonnegative and satisfies the relation
uoL +.
(I+ ~nAn)Un 2 Q-1 + k&n(q * U>, + sn + i&t], Since (q * U),
= Q,U,,
it suffices to choose U, as the positive solution of the linear problem
(I+ r,zAn - kn:nanQn)Un = Un-1 + k&n + &a),
(n = 1,2,. . . ),
IJo=+.
(6.13)
To show that a unique positive solution to (6.13) exists, we observe from Hypothesis (H), (2.2), and (2.16), that the elements ci;) of the matrix C,, G I + rnA, - knu,Q,
possess the properties
c!“.) %l -< 0 for i # j and (n) _ cp qj - 1 + Tn(f-Q,n- bi,, -
b&J - ha,,
j=o
ePjqi,j,n 2 1 - &a,,
j=O
for all i = O,l,. . . , p. This implies that if k,a,, < 1, then C,, is a diagonally dominant matrix and its inverse C;l exists and is a positive matrix (cf. [17]). Hence, a unique positive solution U, to (6.13) exists. Moreover, by the nonnegative property of Qn, this solution can be computed from the iteration process (I + ~-nAn)tJ~~~ = U;_,
+ (k~a,Q,z)U~?
+ k&n
+ &n),
(n = 1,2,. . . ),
(6.14)
uo=e, by the Thomas algorithm, where Uz_i
is the solution of (6.13) at the previous time-step.
In-
deed, since UT, = 0 is a lower solution of (6.13), the well-known monotone convergence theorem
c. v.
876
PA0
(or Theorem 3.1) ensures that the sequence {Ukm”} given by (6.14) with U?’ monotonically from below to a unique positive solution U, (cf. [11,19]). Since
dF!2) $+.n,2)i,n)
= -&%,n,
= 0 converges
1+2) *(%,nr
%,n)
= -bn%,n,
we see from (3.4) and (3.14) that
rt’,‘n= 0, y!2) = bn(q * U)i,n9 2,n where Dn is the maximum of the components Vi,,, of U,. satisfied if (n= k, (a, + b,T’j,) < 1,
A/l(l) =
a,,
Mi2’ =
b,ui,,
(6.15)
This implies that condition (4.11) is 1,2,...).
(6.16)
By Theorem 4.2, we have the following conclusion: under condition (6.16), equation (6.10) (with the boundary and initial conditions in (6.1)) h as a unique finite-difference solution UC 2 0. Moreover, the sequences {Ehm)}, {this’}
obtained from (4.9) with Xii’) = U,,
= 0, I’?’ = diag(#i,. . . , -ygi) converge monotonically where U, is the positive solution of (6.13).
&’
= 0, I$”
to UC and satisfy relation (3.16),
The construction of upper and lower solutions for the above model pro.blems can be used to compute numerical solutions whenever the physical parameters and the boundary-initial functions are given. Some of the numerical results for the special case q*u = u in the above model problems, including some test problems with known analytical solutions, can be found in [14].
REFERENCES 1. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, (1992). 2. J.M. Bebernes and R. Ely, Comparison techniques and the method of lines for a parabolic functional equation, Rocky Mountain J. Math. 12, 723-733, (1982). 3. J. Blat and K.J. Brown, A reaction diffusion system modeling the spread of bacterial infections, Math. M&h. Appl. Sci. 8, 234-246, (1986). 4. T.A. Bronikowski, J.E. Hall and J. Nohel, Quantitative estimates for a nonlinear system of integrodifferential equations arising in reactor dynamics, SIAM J. Math. Anal. 3, 567-588, (1972). 5. V. Capssso, Asymptotic stability for an integrodifferential reaction diffusion system, J. Math. Anal. Appl. 103, 575-588, (1984). 6. W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonlinear Analysis 26, 171-178, (1996). 7. J.J. Levin and J.A. Nohel, On a system of integrodifferential equations occurring in reactor dynamics, J. Math. Mech. 9, 347-368, (1960). 8. P. de Mottoni, E. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, J. Nonlinear Analysis 3, 663-675, (1979). Differential 9. C.W. Cryer, Numerical methods for functional differential equations, In Delay and finctional Equations and Their Applications, (Edited by K. Schmitt), pp. 17-101, (1972). 10. L. Galeone, C. Mastroserio and M. Montrone, Asymptotic stability of the numerical solution for an integrodifferential reaction-diffusion system, Namer. Meth. Part. Diff. Eqs. 5, 76-86, (1989). 11. C.V. Pao, Monotone methods for a finite difference system of reaction diffusion equation with time delay, Computers Math. Applic. 36 (10-12), 37-47, (1998). 12. C.V. Pao, Positive solutions and dynamics of a finite difference reaction diffusion system, Numer. Meth. Part. Diff. Eqs. 9, 285-311, (1993). 13. C.V. Pao, Accelerated monotone iterative methods for finite difference equations of reaction-diffusion, Numer. Math. 79, 261-281, (1998). 14. C.V. Pao, Finite difference reaction diffusion equations with nonlinear boundary conditions, Namer. Meth. Part. Din Eqs. 11, 355-374, (1995). Equations, Academic Press, San Diego, CA, (1992). 15. W.F. Ames, Numerical Methods for Partial Differential
Numerical Methods
877
16. A. Berman and R.L. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, (1979). 17. R.S. Varga, Mutvis Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, (1962). 18. AH.. Mitchell and D.G. Griffiths, The Finite Difference Method in Partial Di@rential Equations, Wiley, New York, (1986). 19. C.V. Pao, Numerical methods for semilinear parabolic equations, SIAM J. Numer. Anal. 24, 24-35, (1987). 20. AS. Householder, The Theory of Matrices and Numerical Analysis, Blaisdell Col., New York, (1964). 21. J.P. Kerneves, Enzyme Mathematics, North Holland, Amsterdam, (1980).
computers 8 mathematics wlth~btiafm
PERGAMONComputers and Mathematics with Applications 41 (2001) 857-877
www.elsevier.nl/locate/camwa
Numerical Met hods for Nonlinear Integro-Parabolic Equations of F’redholm Type c. v.
PA0
Department of Mathematics, North Carolina State University Raleigh, NC 27695-8205, U.S.A. (Received March 8000; revised and accepted August 2000) Abstract-This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem ss well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm. @ 2001 Elsevier Science Ltd. All rights reserved.
Keywords-Integrc-parabolic equation, Finite-difference solution, Reaction-diffusion convection, Monotone iteration, Error estimate, Upper and lower solutions.
1. INTRODUCTION Integro-parabolic
differential
equations
of Fredholm
type arise from various fields of applied sci-
ences and have been investigated
by many researchers
therein).
Most
in the literature
equations
such as existence
dependent
of the discussions
solution
and uniqueness
(cf. [l-S]).
in the field (cf.
are devoted
of a solution
[l] and the references
to qualitative
and asymptotic
analysis
behavior
The work in [9,10] is related to finite-difference
of the
of the time-
solutions
of some
In this paper, we give a numerical treatment for a class of nonlinear integr+parabolic equations of F’redholm type under either linear or nonlinear boundary conditions. This treatment is based on the finite-difference method and the method of upper and lower solutions which lead to some monotone iterative schemes for the computation of numerical solutions as well as error estimates between the true finite-difference solution and the computed iterations. It also yields an existence-uniqueness theorem for the nonlinear finite-difference system.
special
type of integro-parabolic
equations.
0898-1221/01/t - see front matter @ 2001 Elsevier Science Ltd. All rights reserved. PII: SO8981221(00)00325-4
Typeset by -&S-W
c. v. PA0
858
The parabolic boundary-value problem under consideration is given in the form
ut-Duzr+V2Lz=f(Z,t,U,q*‘LL),
-a(“)u,(O, t) a(')u,(L,
+
t) +
(0 <
p(“)u(o,t) = g(O)(t, up,t)), pu(L,t) = gyt, u(L, t)),
<
L, 0 < t 5 T),
(1.1)
(0 < t I T),
4x7 0) = $4x), where D E o(z,t) positive constant,
2
(0 <
2
<
L),
and v E v(z,t) are the diffusion and convection coefficients, T is an arbitrary and g(O)(., u) and g(l)(., U) can be linear or nonlinear functions of u. The
function f(., U, w) is, in general, nonlinear in u and v with w = q * u, where
(q*u)(x,t)
G
IL
q(x,
x’, t)u(x’, t) dx’,
(0 Lx 5 L, 0
I: T).
0
(1.2)
It is assumed that the functions D(z, t), v(x,t), f(x, t,u,v), q(x,x’, t), g(O)(t, u), g(l)(t, u), and $J(z) are all continuous in their respective domains, D(s, t) is strictly positive in Z+ E [0, L] x [0, T], and q(z, z’, t) possesses the property
s L
and
q(x,x’, t) 2 0
q(x, x’, t) dx’ 2 1,
0
((x,t)
E DT).
(I-3)
The coefficients a@), pee) (e = 0,l) in the boundary condition are given either by ~(~1 = 0, pee) = 1 (Dirichlet condition) or by a (0 = 1 and pee) 1 0 (Neumann or Robin condition). In the case of Dirichlet boundary condition, the function gee)(t, u) is taken as g(l)(t) which is independent of u. To discretize problem (1.1) by the finite-difference method, we use the implicit scheme for parabolic equations and approximate (1.1) by a system of nonlinear finite-difference equations similar to that in [ll] for time-delayed parabolic equations. The use of the implicit scheme is critical in obtaining a meaningful finite-difference solution which preserves the qualitative properties of the original solution (see [12,13] and Remark 4.1). The purpose of this paper is to develop various monotone iterative schemes for the computation of the solution of the finitedifference system, including the existence and uniqueness of a finite-difference solution and error estimates of the iterative schemes. The iterative schemes are based on the method of upper and lower solutions which depend on the monotone property of f(., U, V) with respect to w. In these iterative schemes, the iterations are governed by a system of linear equations which can be easily solved by the standard Thomas algorithm. The plan of the paper is as follows. In Section 2, we formulate a finite-difference system of (1.1) by the implicit method and impose a basic hypothesis on the coefficient matrix so that the various types of boundary conditions can be included in the same framework of discussion. A monotone iterative scheme for the finite-difference system is given in Section 3, where the nonlinear function f (. , u, v) is considered nondecreasing in w. Similar monotone iterative schemes are given in Section 4 for some more general classes of nonlinear functions. In Section 5, we obtain an error estimate for each of the monotone iterative schemes. Applications are given to three model problems in the final section to demonstrate some techniques for the construction of upper and lower solutions and the implementation of the computational schemes.
2. THE Let h = Ax = L/M,
FINITE-DIFFERENCE
SYSTEM
Ic, = t, - tn_l, and xi = ih, and let A={(xi,t&
i=O,l,...,
M, n=l,...,
N},
Numerical Methods
859
where M and N are the respective number of divisions in [0, L] and [O,T].
When no con-
fusion arises, we write (i, n) instead of (zi,t*). At each mesh point (zi,tn), Di,, = D(%L)r Vi,n = V(Q, t,), and approximate the integral
we let ~i,~ =
U(Zi,k),
by the finite sum ((Cn) E A),
(2.1)
j=O
where u z (u~,~, . . . , UM+) T denotes a column vector in W”+’ weights possessing the property
and (pj}
is a set of quadrature
M O
51,
for (i, n) E A.
Pj 4i,j,n5 1,
c
(2.2)
j=O
The simplest choice of {pj} (q *u).
is pj = h for all j so that approximation (2.1) is a Riemann sum of
Define f(‘LLi,n,(4 * U)i,n) = f(G,L
U(G,L)r
gCO)(uo,n) = 9(o)(GI, u(O,L)),
(9 * U)(%,L)),
9(1)(Q&n) = 9%-6,
+,
L)).
(2.3)
By using the standard central difference operators
u4,nl =
(2)
&+J
( >
= +
(‘1Li+1,n - %-l,n), (2.4)
(%+1,n- 2ui,n + ui-l,n),
and the implicit method for parabolic equations, we approximate the integro-parabolic differential equation in (1.1) by the finite-difference equation IC,-%i,n -‘Iii ,n-l > -
a4,nl + ~zbi,?J= f(%,n, (q* U)i,n).
(2.5)
A simple computation shows that the above equation for the interior mesh points may be written as (1+2r,Di,n)ui,n =
U&n-1
-r, +
Di,,, + ++I,, K W(Ui,n, (!I * U)i,n>,
+ (Di,n- ++I+] (2.6) (i = 1,...,M-1),
where T,, = k,,/h 2. Similarly, by letting i = 0 and i = M in (2.5) and using the central difference approximation for ~~(0, tn) and u,(L, tn) in the boundary condition of (l.l), we obtain the following finite-difference approximations at the boundary points:
1 (q*4o.n)+r,@Do,n + h~o,n)dO)(uo,n>, = ‘1~0, n--l+ bJ(wLO,nt [
[
1 + 27-,Do,n + r,hP(“)(2Do,,z
1+ 2r,D~,,,
+ rJ@(‘)(2D~
+ hvo,n) uo,n - 2~nDo,nqn
,z - hVM,,J u~,n - 27-nJh4,nw--l,n
(2.7)
I
(see [12,14]). Th e initial condition is approximated by
w,o = %h,
(i = 0, 1, . . . , M),
(24
C. V. PAO
860
where ¢i = ¢ ( x i ) . System (2.6)-(2.8) gives a finite-difference a p p r o x i m a t i o n of the continuous p r o b l e m (1.1). If the b o u n d a r y condition at x = 0 is of the Dirichlet t y p e
u(O,t) = g(°)(t), then the first
equation in (2.7) is replaced by
(l+2rnDl,n)ul,n-r,~(Dl,n
h2'n)u2,n (2.9)
=Ul,n_l+knf(Ul,n,
(q* U)l,n)@rn (Dx,n + ~ ) g ( 0 ) ,
and is used for i = 1 in (2.6), where g(0) = g(O)(tn). Similarly, if the b o u n d a r y condition at x = L is given by u(L, t) = gO)(t), then the second equation in (2.7) is replaced by
(l+2rnDM_l,n)UM_,,n--rn(DM-l,n+hVM2--1'n)UM-2,
n (2.1o)
= U M - I , n - 1 + k n f ( U M - l , n , (q * U)M-1, n) + rn
DM-I,n
and is used for M - 1 in (2.6), where g(1) = g(1)(tn). In this situation, the finite-difference s y s t e m consists of M or M - 1 equations depending on whether the Dirichlet b o u n d a r y condition occurs in one or b o t h b o u n d a r y points. To include the various possible combination of b o u n d a r y conditions and some other possible forms of equations (such as equations in polar form) in the same framework of discussion, we consider a system of (p + 1) equations in the form (1 + rnai,n)Ui,n --
rn(b~,nUi-l,n + b~,nUi+l,n) = ui, n-1 + knf(ui,n, (q * u)i,n), (i = 1 , . . . , p -
1),
u (l+rnao,n)Uo,n-rnbto,nUl,n=Uo, n_l+knf(uo,n, (q*u)o,n)+rn bog (0) (o,n), (2.11) (1 + rnap,n)Up,n - rnbp,nUp-1, n = up, n-1 + knf(Up,n, (q * U)p,n) + rnb~g(1)(Up,n), ui,o=¢~, ( i = O , l , . . . p ) , where ai,n, hi,n, and b~,n are positive functions satisfying ai,n _> bi,n + b~,n and p stands for 1, or M - 2 depending on the type of b o u n d a r y conditions at x = 0 and x = L. Specifically, p = M if a (°) = a (1) = 1, p = M - 1 if a (°) = 1, a (1) = 0 (or a (°) --- 0, a (1) = 1), and p = M - 2 if a (°) = a(1) = 0. (In the latter case, the vector un defined below should be M,
M
-
un =
For each n = 1, 2 , . . . , define column vectors in R p+l by Un = (U0,n, Ul,n,... , Up,n) T,
Vn = (VO,n, V l , n , . . . , Vp,n) T,
(q * u)~ = ((q * u)0,n, (q * U ) l , ~ , . . . , (q * u)p,n) T,
(2.12)
F(un, Vn) = (ro(UO,n, v0,n), Fl(Ul,n, Vl,n),..., Fp(up,n, Vp,n)) T, where (q * u)i,n, i = 0, 1 , . . . ,p, are given by (2.1) (with M = p), and
Fi(Ui,n,Vi,n) = f(Ui,n,Vi,n),
(i = 1,... ,p -- 1),
Fo(uo,n, Vo,n) = f(uo,~, VO,n) + rnbog(°)(uo,,~), p,n) =
+
(2.13)
•
Define also a tridiagonal m a t r i x An by
ao,n -bl,n
b'
0
- - O~n
al,n "..
An =
• .,
"..
(2.14)
". ".
0
-bp,,~
'
--
b!p - - l ,
ap,n
Numerical Methods
861
Then we may write system (2.11) in the compact form
(I + rnAn)un = Un-1 + k n f ( u n , (q * u)n),
(n = 1, 2 , . . . ),
uo = ¢ ,
(2.15)
where ¢ = ( ¢ 0 , . . . , Cp)T. Notice from (2.1) that we may write (q* U)n = Qnun,
withQn=(pjqi,j,n),
i , j = O , 1,...,p.
(2.16)
It is clear from (1.3) and (2.2) that Qn is a nonnegative matrix and the sum of the elements of each row of Qn is bounded by one. To compute the solution of the finite-difference system (2.6)-(2.8), it suffices to do the same for system (2.15) with suitable conditions on An and F(un, vn). We impose the following basic hypothesis. HYPOTHESIS ( n ) . (i) The elements of An possess the property
bi,n>O,
b~,n>O,
and ai,n>_bi,n+b~,n,
((i,n) E h ) .
(2.17)
(ii) F ( u n , Vn) iS a Cl-function Of Un, Y n in a subset Sn O[R p+I. It is easy to see that by choosing the spatial increment h satisfying the relation h < 2 \lv~,nl] '
((i,n) c A),
(2.18)
condition (2.17) is satisfied by the coefficients in system (2.6),(2.7). In particular, condition (2.18) holds for every h > 0 if there is no convection (that is, v~,n -- 0). On the other hand, if the effect of convection dominates diffusion to the extent that it may require a prohibitively small h (that is, D~,n/v~,n is extremely small) an upwind differencing scheme leads to condition (2.17) without any restriction on h (cf. [12,15]). Hence, condition (2.17) can always be satisfied by system (2.6)-(2.8), including the Dirichlet boundary condition in (2.9) and (2.10). The hypothesis on F (un, vn) is also satisfied when f (., u, v), g(0) (., u), and g(1) (., u) are continuously differentiable in u and v. Condition (2.17) ensures that the matrix An is an M-matrix and for any rn > 0, the inverse matrix (I + mAn) -1 exists and is a positive matrix (cf. [16,17]). Moreover, the smallest eigenvalue #n of An is real and nonnegative, and if for each n, the strict inequality
ai,n > bi,n + b~,n holds for at least one i,
(2.19)
then #n > 0 (cf. [17]). The subset Sn in Hypothesis (ii) is taken as the sector between a pair of upper and lower solutions whose definition depends on the monotone property of F(un, Vn) with respect to vn (see (3.2) in Section 3). This property and the corresponding definition of upper and lower solutions are given in the following two sections. 3. M O N O T O N E ITERATION QUASI-MONOTONE NONDECREASING
FOR FUNCTIONS
To develop monotone iterative schemes for the finite-difference system (2.15), we require some quasi-monotone property of the function F(un, vn) with respect to Vn (but not with respect to Un). Recall that F(un, Vn) is said to be quasi-monotone nondecreasing (respectively, nonincreasing) in $n if ~v" > 0 (respectively, ~ < 0) for each i = 0, 1,... ,p and each (Un, vn) in $n, where Fi =- F~(ui,n,vi,n) is given by (2.13) and ~ v - ov,,,~" oF~ In this section, we present a monotone iterative scheme for quasi-monotone nondecreasing functions using either an upper solution or a lower solution as the initial iteration. The definition of upper and lower solutions for the present case is given by the following.
c. v. PA0
862
DEFINITION 3.1. Let F(u,,v~)
be nondecreasing in v, for (un,vn)
E S,.
Then two vectors
iin, ti, in IW’+’ are called ordered upper and lower solutions of (2.15) if ii,, > ti,, and if (I+
r,&)f.&
2 %-i
+ W’(k, (n=
co > +,
(q * ti)n),
(3.1)
1,2,...),
and ti, satisfies the above inequalities in reversed order. The subset S, in the above definition is given by s, = { (un,vn)
E lRp+l x wp+l; ti, I un < tin, (q* ii)n I v, I (q* ti)n}.
(3.2)
For notational convenience, we set sJ&i’= {Un E lwp+i; ti, I lln I Gin}, s$
= {Vn E lwp+l; (q*ql
Iv,
Si,, = { (%n >%,n) E Iw x R
(3.3)
I (q*fi)n},
G,n I %,n I f-k,,, (4 * G)i,n I G,n 5 (4 * ii),,,},
where &,, and tii,n are the respective components of ii,, and ti,. function satisfying
Let Ti,n be any nonnegative
(3.4) and define matrices A, and I,, by A, = I + rnAn + k,J,,
rn = diag(yo,,, ~1,~~. . . , -yp,d
(3.5)
Then problem (2.15) may be written as Au,
= u,-1 + ICn[r,u, + F(u,,
(q * &)I,
(n=
1,2,...), (3.6)
us = $. Relation (3.4) implies that for every v, E SA2’, rnUn + F(u~, v,) L r,u:,
+ F(u;, v*),
when iii, > un 2 uk > i.&.
(3.7)
Moreover, by the nonnegative property of r, and Hypothesis (H), the inverse matrix
AZ1 exists and is a positive matrix (cf. [16,17]). Th’IS ensures that for any initial iteration un(0) , we can construct a sequence {uL~’ } from the linear iteration process A&“)
= u;_i
+ k,
[I’,ui”+‘)
up=q/J,(n = 1,2,. . . ),
+ F
(uim-‘),
(q * u);~-‘)
>I,
(3.8)
is the solution of (2.15) (or (3.6)) at the previous time step. It is wherem=1,2,...,anduA_i obvious that this sequence is well defined and can be easily computed by the Thomas algorithm because &
is a tridiagonal matrix (cf. [15,18]). Denote the sequence by {Eim)} if u$? = ti,,,
and by {ail”‘}
if u?’
= h,, and refer to them as maximal and minimal sequence, respectively.
We first show that if u;_i E St!, and is a solution of (2.15) at n - 1, then the maximal and minimal sequences possess the monotone property i& < r.&) < u(m+l) 5 iiim+i) 5 Eim) < - tin, - --n
m,n=
1,2 ,...,
(3.9)
and their respective limits lim aim) = 8,, m-WC
lim rain’ = By m-CO
(3.10)
exist and satisfy the equations (I+
r,&)u,
=
n;-1 + W’(u,,
uc = @.
(q * u),),
(n=1,2,...),
(3.11)
Numerical Methods
863
LEMMA 3.1. Let F(un, v,) be nondecreasing in v, for (un,v,J
E S,,, and let u;_~
E St?,
with UT, = 111. Then the sequences {iI&~‘}, {gim’ } governed by (3.8) possess the monotone property (3.9). Moreover, the limits iin and gn in (3.10) exist and satisfy (3.11). I& in addition, u; E S,$” and is a solution of (2.15), then gn I u; I i&.
PROOF. Letv&? = ii?)-ii:), wherei$ = iin. By (3.1), (3.5), and (3.8) (with m = l),
A,V(o) > tii,_l n = i&-l
-
u;_~+ k,
- u;_1,
(iin -ii;) >+
I’, [
F(ii,,
(q * a),) - F
(Tit’,(q * ix)~‘)]
(72 = 1,2,. . . ),
and w$@ = C-0- q!~2 0. Since by hypothesis iii,_1 2 u;_~, the positivity of A;ll implies that 2 0. This leads to ii:)
wi”)
gives &’ F(w,
v,)
2 L&?, where &’
2 EL’). A similar argument using the property of a lower solution = ii,.
Moreover, by (3.8),(3.7) and the nondecreasing property of
in v,, we have
which yields iii’) 2 gc’. This shows that --(O) u, > _ iii’) > &’ 2 &?. An induction argument, using property (3.7), leads to the monotone property (3.9). It follows from this property that the limits -& and gn in (3.10) exist and are in Si”. Letting m + 00 in (3.8) and using the equivalence relation between (3.6) and (2.15), show that tin and gn satisfy equation (3.11). To show the relation iin 2 ui when u: is a solution of (2.15), we observe that iiim’ = ut; = + and u: satisfies the equation dnu;: = u;-~
(n=
+ kn[I?nu; + F (u;, (q * ~*),a)],
1,2,...).
(3.12)
Substraction of the above equation from (3.8) yields
A,
(ii$$ -u*
n> =k,
[r, (~$‘+‘)-u$tF
(Bim--l), (q+E):m-l))-F(u;,
(q* u*),)]
, (3.13)
m = 1,2,. . . .
Consider the case m = 1. Since YiF) = tii, 2 u:, relation (3.7) and the nondecreasing property of F(u,, v,) in v, imply that &(iig) - uz) 2 0. This leads to ii;‘) 2 u:. Assume, by induction, that ail-‘) 2 uz. Then by (3.13),(3.7) and the nondecreasing property of F(u,,vn) in v,, we obtain &(Eim) - LIE) 2 0. This gives aim) 2 u;t. It follows from the induction principle that irim) 2 u; for every m. Letting m + 0;) and using (3.10) show that Q 2 u;. A similar argument gives ui 2 gn. This proves the lemma. I To show the relation iin = fin (= u;) and u: is the unique solution of (2.15), we set u n=max
{
iW,=max
$+,V&
(‘LLi,n,%,n)E %a,
i=O,l,...,
p
) >
2(%,&J
/ ; (W,n,‘Ui,n) E %I,
{I
(3.14)
i=O,l,...,p. 1
The following theorem gives the monotone convergence of the maximal and minimal sequences.
THEOREM 3.1.Let iin, tin be ordered upper and lower solutions of (2.15) where F(u,, v,) nondecreasing in v, for (u,, v,) E S, . Assume that Hypothesis (H) and the condition
Man + Mn) < 1,
(n=
1,2,...)
is
(3.15)
c. v.
864
PA0
are satisfied. Then a unique solution ui to (2.15) exists and is in S,(I). Moreover, the sequences {G
%${GP}
g’lven by (3.8) converge monotonically to u: and satisfy the relation
G, 5 &ml < u(m+l) 5 u:, < Eim+l) -< Eim) -< iin, - --n PROOF.
Assume, for the moment, that UC-~ E St?,
{aim)}, {ail’}
(m, n=
1,2 )... ).
(3.16)
with uz = $. By Lemma 3.1, the sequences
converge monotonically to their respective limits &
and u, that satisfy (3.11).
This implies that w, E iii, - u, satisfies the relation (I + r,A,)w,
= WY%
(q * %)
- F(gn,
(q * &)I.
(3.17)
Define the Jacobi matrices F,(u,,v,)
~Fo
z diag
-(
DUO,,
a&J
)I7 - (Up,nr 9.
UO,n,uo,n) 7..., (%,n7 1JP,?a aUP,n 8FP
'UP,73
dVP,n
By the mean-value theorem and (2.16), equation (3.17) may be written as (I + m&)wn
= k-&(&z,
~ln)w, + &(Cn, rln)(s * w)nl
= k&(&z,
rln) + F,(E,,
w-JQn]wv
.(3.18 (3.19)
where (&, vn) is an intermediate value in S,. Define P, = I+r,,A,
-kn[Fu.(Sn,rln)
+Fu(&z,rln)Qnl.
(3.20)
Then relation (3.19) is equivalent to P,w,
= 0,
(n=
1,2,...).
(3.21)
We show that under condition (3.15) the inverse matrix PC’ exists and is a positive matrix. Let P
= (pi”‘) *,n 14.7.
p, and let the diagonal elements of F,,(<,, 77,) be denoted = O,l,..., By (2.16) and the nonnegative property of F,,([,,qn), the elements of are given by d,,,(pj qa,j,d and are all nonnegative. In view of (3.20)
by do.n,k,n, . . . , dp,w the matrix 4,(L,vn)Qn and (2.14), the off-diagonal elements pi:) of P, possess the nonpositive property p$) < 0 when i # j. Moreover, by (3.14), p(n) x,2 > 1+?-,a. 2,n - k(o,
+ M,z . pi ~i.i,n),
j#i
j#i
It follows from Hypothesis (H) and conditions (2.2) and (3.15) that P
2 1 - Ic,(a,
+ M,)
> 0.
This implies that ply) > 0, ply) < 0 if j # i and P, is a strictly diagonally dominant matrix. It followed from these properties that P;’ exists and is a positive matrix (cf. [16,17]). Hence, w = 0 which shows that iin = gn. To complete the proof of the theorem, we show the existence of a unique solution by a ladder argument, starting from n = 1. Since UT,= + E Sil), Lemma 3.1 ensures that the sequences {-ii\“‘}, {Gil”‘} converge monotonically to their respective limits 31 and u, that satisfy (3.11) when n = 1. Lemma 3.1 and the above argument for n = 1 shows that El = ul and their common value is the unique solution of (2.15) at n = 1 and is denoted by UT. Knowing the solution u; and u; E Sill, the same reasoning shows that the sequences {iiim,“‘}, {u$~“‘} converge monotonically to a unique solution u; and u$ E Si’). to the conclusion of the theorem.
A continuation of the same process leads I
Numerical Methods
4. MORE
GENERAL
When the function F(u,,
v,)
865
QUASI-MONOTONE
FUNCTIONS
is quasi-monotone nonincreasing, the requirement of upper and
lower solutions is coupled and is given in the following.
DEFINITION 4.1. Let F(unrvn) be nonincreasing in v, for (un,vn) E S,. &, 6, are called coupled upper and lower solutions of (2.15) if G,, 2 G, and (I + T&J%
2 G-1
+ kJ(k,
(I + ~,x&)iL, I %,--I + kF(&,
Then two vectors
(q * cl),), (n=
(q * fi),),
1,2,...),
(4.1)
tie > + 1 tie. The subset S, in the above definition is given by (3.2) with respect to the above fin and ti,. Notice that the requirement of &, ti, in (4.1) is coupled, which leads to the following inter-related (but uncoupled) iteration process: A,&“)
= u;_~ + k, ~J%zcm-‘) + F @+l),
(q * fi)im-“)]
d,gkm)
= u;_~ + Ic, l?,&+‘) C
(q*E)im-‘)
E;m) = dm’ where u;_~
= $
m=
7
+ F
(
I&~-“,
1,2,...,
,
>I
,
(4.2)
(n=1,2,...),
is the unique solution of (2.15) in S:!,.
It is clear from Hypothesis (H) that
the sequences {YiLm)) , {I&~’ ) are well defined and can be computed by the Thomas algorithm whenever u;._~ is known. We show, as in Lemma 3.1, that these sequences possess the monotone property (3.9) and their limits _iin,gn in (3.10) satisfy the relation
(4.3) T&-J=L&
LEMMA 4.1. Let F(un,vn) with
UT, = +.
(n = 1,2,. . . ).
=+,
be nonincreasing in v,
Then the sequences
{Tiim)},
for (un,vn)
u in (3.10) satisfy relation property (3.9), and their limits ii,, _~ and is a solution of (2.15), then iin 2 u*71--T%’ > u PROOF. Let %Vi”)= iiF) -ii?)
and &’
= g?’
E S,,
and let u;_~
E St?,
{gimm,} governed by (4.2) possess
-&‘,
(4.3).
the monotone 16 in addition, I$ E S, (1)
where ii(‘) = tii, and gi”’ = ti,. By (3.5),
(4.1), and (4.2), we have -(O) w. = Co - $ 2 0, s (‘) = $ - ii0 >“o and d,%(0)n
2 (ti,_,
- u;_~)
+ k,
[
l?,
(
tin -iii’)
)
+ F (tin, (q * %)
- F @ho’, (q * u):‘)]
= tin-1 - u;_1 2 0, -&x,z to) _> (u;t_,
-i&-l)
+ k, [r,, (I& - fin) + F (,:‘,
= tin_1 - u;_1 2 0,
(q * ii(o))n)
- F (fin, (q * ti),,)]
(TX= 1,2,. . .).
By the positivity of Ail, i%$) 2 0, and vvi” 2 0 which yield ii:) 2 T&l) and &’ Similarly, by (4.2),(3.7) and the nonincreasing property of F(u,,v~) in v,, A, (Tit’ - &)) and@-&) &)
= k, [r” (@’
- I$‘)
+ F (Ti;‘,
(q * #)
- F (&“,
(q * ii):‘>]
1 &‘.
2 0,
= + - + = 0. This relation and the above conclusion lead to ?rF) 2 iii’) 2 > u(O) ,.... An induction argument shows that the sequences {i@)}, - _n foreveryn=0,1,2
c. v. PA0
866 {I&~)}
possess the monotone property (3.9). This implies that the limits -&,LI~ in (3.10) exist
and satisfy relation (4.3). Now if ut is a solution of (2.15) in Si”,
then UT, satisfies (3.12).
A subtraction of (3.12)
from (4.2) gives the relation A,,
($“‘-f)
=kn
sz, (upQQ) =I& +)
[rn(ti~m-l~-u;) ++in-‘),
(q * g)i@))-F(U;,
[rn(u;-lly,>
u*)&F(&-,
= Am’ = u;, =
+lqu;, m=1,2
$,
(q*
(..‘,
Consider the csse m = 1. Since EF) = tin 2 u: and II:’ the nonincreasing property of F(unrvn) in v, that A, (Ii?) - u;) This leads to Zi,‘) 2 uz 2 r&l’.
2 0
and
(n=1,2
(q *
U*)n)] ,
(q*li)g-‘))I , (4.4) )... ).
- i& I uz, we see from (4.4),(3.7) and
A, (u; - II:‘>
2 0.
An induction argument using relations (4.4),(3.7),
nonincreasing property of F(u,, v,) in vn, yields i@) 2 ui m -+ co shows that & 2 ui 2 LIP. This proves the lemma.
(m) for every m. 2 II~
and the Letting I
Using the functions cr, and M, given by (3.14) with respect to the present coupled upper and lower solutions, we have the following analogous convergence theorem as that in Theorem 3.1. THEOREM 4.1. Let iii,, tin be coupled upper and lower solutions of (2.15) where F(un, v,) is . . nonincreasmg In v,, and let Hypothesis (H) and condition (3.15) be satisfied. Then a unique solution ult, to (2.15) exists and is in Si”. Moreover, the sequences by (4.2) converge monotonically to u; and satisfy relation (3.16).
{ii~m)},{~k~,m)} governed
{gi”‘} converge PROOF. In view of UC;= @ E Si” , Lemma 4.1 implies that the sequences {Ei”‘), monotonically to their respective limits El and II, that satisfy (4.3) at n = 1. Let WI = Ii1 - II,. By (4.3) and the mean-value theorem,
(I + ~lAl)wl = h[J’(%, (q * 41 - WI, (q *WI = h[&(&,m) + (--~d~~,m)Ql)lw. Since the above relation is in the same form as that in (3.19) (with n = 1) where F,(&,ql) is replaced by (-F,(
The I
The method for quasi-monotone nondecreasing and quasi-monotone nonincreasing functions can be extended to a more general class of functions in the form
qu,,
v,) =
F(‘)(u,, v,) + F(2+hvn),
(4.5)
where F(l)( II,, v,) is nondecreasing in v, and Fc2)( u,, v,) is nonincreasing in v, for (u,, v,) E s,. The subset S,, is again given by the sector between a pair of coupled upper and lower solutions which are defined as follows.
NumericalMethods DEFINITION 4.2.
867
are called coupled
be given by (4.5). Then two vectors i&,ii,
Let F(un,vn)
upper and lower solutions of (2.15) if ii,, > iin and
(I + rnA,)tin
2 tin-1 + k,, [F(l) (ii,, (q * a),) + F(‘) (fin, (q * ti)n)] ,
(I + r,&)ti,
5 L-1
(4.6)
+ k, [F(l) (iin, (q * ti)n) + Ft2) (tin, (q * ti)n)] , (n=
Go r + r Go,
1,2,...).
We again define the sectors S,, SLe’, and Si,,, e = 1,2, by (3.2) and (3.3) with respect to the above pair of upper and lower solutions. For each e = 1,2, define yz’z, oie), and Mie’ by (3.4) and (3.14) where Fi(ui,+, vi,,,) is replaced by F!e)(~i,n, Q+). z
The functions y,!,:, r,!“,’ are chosen
such that ~j,z + yj,z 2 0. Define also A, = I + T,A, + k, (I’~‘) + I$))
I’$
,
= diag (-$A,.
. . ,$A)
(e = 1,2).
,
(4.7)
Then by Hypothesis (H), the inverse A,1 exists and is a positive matrix (cf. [17]). Moreover, for every v, E SL2’, I’f)u,
+ FCe)(u,, vn) 1 rr)uk
Using ~2) = tin and &” {irim)}, {&?}
+ FCe) (uh, v,) ,
when tin 2 un 2 uk L tin,
(a = 1,2).
(4.8)
= t, as a pair of coupled initial iterations, we construct two sequences
from the linear iteration process
J&m)
I:) + IF) ~im-1) + ~(1) i$+i), > ( K +1”(Z) Ehm-11, (q * u)C$-r’ (
= u;_l
(q * i-j)im-i)
+ Ic,
)I
A,Llim) =u;_1 +k,
[ (rg) + ri2))&m-l)+ F(l) (&+l),
gg4, (q* qp-1) ( )I $4 = @’ = $ 1 m= 1,2,...,
>
(q* g)$yl’>
(4.9)
+Ja
(n=
1,2,...),
where u; = $J. It is clear that these two sequences are well defined and can be computed by the Thomas algorithm. In the following lemma, we show the monotone property of these sequences. LEMMA 4.2.
be given by (4.5), and let u;_~ E St?,
Let F(u,,v,)
with ug = $J. Then the
sequences {ii~m)},{~&m”’} governed by (4.9) possess the monotone property (3.9). PROOF Let F(O) = iji”) - ii(l) do) = uil) - g:‘,
where ii(O) Tl = fi VIand
and I’* 71= j$,‘) + rr’,
I&” = a 7L*By (n4.6), (4.7), annd;47-d;, we have &w(O)n > - fin_1 - u;_~ + k,
[
r:, ( tin- iiF))
+ FC2) (ii 7LI(q * G),) - FC2) (fi?‘,
+ F(l) (ii,,
(q * a),)
- F(l) (ii?‘,
(q * i?)?‘)
(q * E&c’)]
= tin-i - I$&_, 2 0, d,vvi’)
> - u;_i
- i&-r+
+ Fc2) (@, = u;_l
-ti,_i
The positivity of A,1
k,
[
r:, ( lly -
(q * n)?)) 10,
^ un) + F(l) ( a;),
- FC2) (L (n=
implies that Vt)
(q * s)p’)
- F(l) (tin, (q * ti)n)
(q * a),)]
1,2,...).
2 0 and v&c) 1 0 for n = 1,2,. . . . It is obvious that
W(O) - 0 and w-, = $ - tia 1 0. This shows that i$) 7z = tic - Q >
5 @‘I
and &’
5 ~2)
for
868
c. v. PA0
n = 0, 1,2, . . . . Moreover, by (4.8),(4.9) have ZF’ = &)
and the monotone property of Fte)(un, v,)
in v,,
we
= $J and
A, (i%(l) -Q)
= k, p;
(@
+Fc2) (Ii;), This leads to ii:)
2 gc’
-g?‘)
+ F(i) (ii?‘,
(q * I#)
(a * Q’)
- F(l) (ll?,
- Fc2) (I$“, (qci):))]
(q * t&c’)
2 0.
for n = 0, 1,2, . . . . The monotone property (3.9) follows by an induction
argument.
I
Because of the monotone property (3.9), the limits iin and gn in (3.10) exist and satisfy the relation ti, 15 gn I iin I tin. Letting m + co in (4.9) and using relation (4.7) show that E~ and an satisfy the equations (I + r,A,)En
= u:-~ + k, [F(‘)(G,
(q * a),) + F’2’(Ti;2, (q * a)n)] ,
(I + ~,A,)~~
= u;_*L-_I + k, [J’%n,
(q * u)n) + F(2)(~nr
Es = L& = ?j$
(n=
In the following theorem, we show that En = u,(=
(q * i&z,]
(4.10)
,
1,2,...).
uz) and u: is the unique solution of (2.15).
THEOREM 4.2. Let F(u,,v,J be given by (4.5), and let iii,, ii,, be coupled upper and lower solutions of (2.15). Assume that Hypothesis (H) and the condition k, (o$” + ,i2) + Mi1) + A4i2)) < 1,
(n = 1,2,. . . )
are satisfied, where CL” and MAe’, .C = 1,2, are given by (3.14) with respect Then a unique solution {xim’}
II; to (2.15) exists and is in SA”.
(4.11) to Fce)(un,v,).
Moreover, the sequences {iiim)},
governed by (4.9) converge monotonically to UC and satisfy
(3.16).
PROOF. Consider the case n = 1. Since UT,= $ E S,$‘), Lemma 4.2 ensures that the sequences {a’;“‘},
{ui”‘}
possess the monotone property (3.9), and therefore, the limits ili,u,
exist and satisfy (4.10) at n = 1. Let w$l) = iii’) -gi’). mean-value theorem.
(I+ n&b,
(l) = ICI [F(‘)(i&, (q*ii)l) - F(%,, +F(2)(&, (q* u)d - Fc2h,
in (3.10)
Then by (4.10) (with n = 1) and the
(q * 41) (q * n),)]
where (Cl, ~1) and (G, 77:) are some intermediate values in Si. Since the above relation is the same as that in (3.19) (with n = 1) when F, and F, are replaced, respectively, by Fil’ + Fi2’ and F$‘) - Fi2), and since Fil’(&,
T.II)+ Fi2’ (& 77’) I ai’) + a?),
IF;‘)(Em)
- FJ2’ (S;,s;)l
5 M,(l) + M,(2),
we conclude from the argument in the proof of Theorem 3.1 and condition (4.11) that iii = a,(= UT) and UT is the unique solution of (2.15) at n = 1. Moreover, relation (3.16) holds at n = 1. Knowing the solution u; and UT E Sil) the same ladder argument as that in the proof of Theorem 3.1, using the conclusions in Lemma 4.2, shows that a.unique solution ui E SA” to (2.15) exists and the sequences {YiAm)}, {@“‘} relation (3.16). This proves the theorem.
converge monotonically
to UC and satisfy I
NumericalMethods
869
REMARK 4.1. (a) When f(z, u, V) = f(z, U) is independent of v, the definitions of ordered and coupled upper and lower solutions coincide, and problem (2.15) is reduced to a finite-difference system for standard parabolic boundary-value problems. Hence, the results in Theorems 3.1, 4.1, and 4.2 are directly applicable to this class of parabolic equations (cf. [13,19]). (b) In the formulation of the finite-difference system (2.6)-(2.8), we have used the implicit method for parabolic equations.
This approach ensures that the computation
of the
iterations aim) and girn’ from any one of the linear iterative schemes (3.8), (4.2), and (4.9) is unconditionally stable with respect to the mesh sizes k, and h (cf. [15,18]). This fact and the consideration of the variable time-increment k, make it more attractive in the computation of numerical solutions with large time t,, especially in relation to the asymptotic behavior of the solution. It should be pointed out that if the explicit or a semi-implicit method is used, then the resulting numerical solution may be incorrect, misleading, or incompatible with the solution of the continuous problem. Some examples for this situation can be found in [13,14].
5. ERROR
ESTIMATES
AND
NUMERICAL
STABILITY
It is seen from the monotone convergence of the maximal and minimal sequences {x;~)}, {L&~)} that given any EO> 0 and any norm in lfUP+l,there exists an integer m* = m*(co) such that lliml - u:, < when m 1 mm, - iiim) -l&rim’ < EO, I/ II II I/ where uimn,stands for either ai*) u:
and the mth approximation
or zLm’. This shows that the error between the true solution L&“’ is bounded by EOwhen m 2 m*.
computation, the true solution ui_i
However, in actual
in the iteration process is taken as ui:‘i
for some m’. This leads to an error between the theoretical and the computed mth approximation for each m and each n. In addition, there may be round-off error and possibly initial error in the computation of L&~). The aim of this section is to give an error estimate between the true solution uz and the computed mth approximation vvimm,for each of the iterative schemes (3.8), (4.2), and (4.9). Since for any norm in Rpfl ,
the convergence of {@“‘}
to ui implies that
wp - u:,II<- IIwp - up) II+
E,,
when m 2 m* .
(5.1)
Hence, to investigate the error between the computed mth approximation whrn”’ and the true solution ufi, it suffices to obtain an error estimate between wnCm)and uLmm, for a suitably large m 2 m* . We do this for each of the three types of quasi-monotone functions F(u,, v,) treated in the previous two sections. For technical reasons, we assume that & = A and k, E k are independent of n, and choose ^fi+ = 7 so that I’, E l? = yl and A, = A. 5.1. Quasi-Monotone
Nondecreasing
Functions
For quasi-monotone
nondecressing functions, the iteration process is given by (3.8) and the computed mth iteration w,,(m) satisfies the equation dwim) = u;__~ + k I’w(“+~) +F 71
w(“+ It
(q * w)(“‘--~) n
>I
+ eim),
(5.2)
c. v.
870
PA0
where WV;‘) is either iin or ti, and eirn’ denotes the combined error due to round-off and the approximation of uz_. 1 by w;~‘). Let e, and e, be the maximal round-off error and the maximal allowable error between the theoretical and computed mth approximation, respectively. Then by the convergence of the maximal and minimal sequences in Theorem 3.1, there exists an integer m* such that ei? I/
II
when m 2 m*,
(n=
1,2 I...) N).
(5.3)
Let [lQnll denote the operator norm of the matrix Q,, and define (5.4) where y, u,, and ikf, are given by (3.4) (with y > ~i,~) and (3.14), respectively, and p > 0 is the smallest eigenvalue of A. In terms of the parameter wn, we have the following error estimate between the true solution and the computed approximations. THEOREM 5.1.
Let II; be the true solution of (2.15) where F(unr vn) is nondecreasing in v,, and
n = ti,. Assume let {w;~“‘} be th e computed sequences from (3.8) with either w,,(‘) = ti n or w(O) that Hypothesis (H) and condition (3.15) are satisfied. Then there exist an integer m* and a norm in KY’+’ such that
II,pm, - II e,, u;
< - 1-
kw,
for all m 2 m*,
(72 = 1,2,. . . ) Iv),
(5.5)
provided that kw, < 1. PROOF. In view of (5.1),
it suffices to obtain
an error estimate
for I[w~~,“’- uL~) 11 with
m > m* and a sufficiently small E,, where ui”’ = who’. Let Ehm’ = wimLm, - uimm,and IELm’I , where E!“) 2,71) i=O,l,..., = (pm)) p, are the components of Ekm’. By subtract0,n ) ... 7IJ$?I)T 1 ing (3.8) from (5.2) and using the mean-value theorem, we obtain
w;“+, - F
(uim-‘),
=k [I?+ F,(&,vn)
(q *
u)$-
+ F,(&x,
(q * w)$‘+l)
>I+ eim)
>
wdQn1J$‘+l) + ek’?
where (&, q,) is an intermediate value. In view of (3.14), y + ~7~2 0, and the positivity of d-l and Qn, the above relation yields the estimate
lEim)) I A-’ {k [(y + a,)1 + I&Q,]
IEhm-l)l + ieim’i},
(5.6)
and etm), i = 0 1 .., p t are the components of eim,“‘. Since where je?)I = (Je(m)I 0,n i ...) leimn)I)T I the spectral radius of A-’ is bounded $/3-l, whek /3 is given by (5.4), we see from (5.3) and (5.4) that given any small 6 > 0, there exists a norm in IF’+’ such that (5.7) for all m 2 m*, where IIQnII is the operator norm of Qn (cf. [20]). Let b, = kw,, E = e, + e,, and choose E sufficiently small, where w, is given by (5.4). Since ,8-l 5 1, the above relation yields l/E~~)II~b,I/~~~-‘)lj+e, In view of EL’) = wi”’ - I$’
whenm>m*,
(n=1,2,...,~).
(5.8)
= 0, an induction argument in m (with n fixed) gives
IJE$-~)I/< (~~~-1)+b~~-2)+...+1)~<(1-b,)-1~,
(m
The above inequality and relation (5.1) (with EOsufficiently small) leads to estimate (5.5). proves the theorem.
This I
NumericalMethods
5.2. Q&G-Monotone
Nor&creasing
For quasi-monotone
871
Functions
nonincreasing functions, the computed mth iterations &n,“) and P&~’
satisfy the equations A%$$,“) = u;_i
+ k l?i+@)
[
+ F Wim-l), (q * w);y))] (
+ i$‘, (5.9)
w(“+l), (q * .iiiT)im-l)_ + girn), AELm) = u;_~+ k 13vim-‘) + F (_?z [ >I
where CZirn)and TV”’ denote th e combined round-off and allowable errors between aim) and i~$~) and between r&’ and y&?, respectively. By the monotone convergence result of Theorem 4.1, there exists an integer m* such that Il~~~)ll+Il~~m)ll
forallm2m*,
(5.10)
(n=1,2,...,N).
In the following theorem, we give an analogous estimate as that in Theorem 5.1. THEOREM 5.2.
Let UC be the solution of (2.15) where F(h,vn)
is nonincreasing in v,,
and = tii, and yip’ = ti,.
let {iVim)} and {ail’}
be the computed sequences from (4.2) with w?) Assume that Hypothesis (H) and condition (3.15) are satisfied. Then there exist an integer m* and a norm in lRP+l such that
IImy-,;II+II&Lu;II
(n=1,2
forallm>m*,
l-kw,
,...,
N),
(5.11)
provided that lcw, < 1. PROOF. Let ELm’ = dnm) - grn) and B,“’ = ail’ applying the mean-value theorem, we obtain
d$?
= k ki$m-l)
+ F,(&,qn)$~-‘)
A@’
= k rEi+
+ F, (<;, y/;)&+‘)
where (L,s)
[
and (C,,$,) 1E;“‘l
Id-‘{
Let Eim) = Ekm’ +&ml
-ghm’.
By subtracting (4.2) from (5.9) and
+ F&,
,n)Q,&m-l)]
+ F,, (t;, q;) Q,&&‘]
+ z$,~‘, + e,“‘,
are some intermediate values. The above relation implies that
(~~~-l'l+M,Q,I~~~-l)I] +I+'I},
k[(r+cm)
and IeLm’1= Ieim)I + I@’
I. Then addition of the above equations leads
to
IIE~~)/ I A-'{ [k(y+c,)I+
ACLQ,](~;y+~)l+ leim)l}.
(5.12)
Since the above inequality is of the same form as that in (5.6), we conclude from (5.10) and the argument in the proof of Theorem 5.1 that (5.11) holds. I 5.3. The Extended If the function F(h, the relation
Quasi-Monotone
Functions
v,) is given by (4.5), the computed mth approximations ~v:~),v&~) satisfy + k r’+$+‘) + j’(i) +m-‘), [ ( (q * w)(m-i) +F@) ++‘), T& ( ?a
&+-km’ = u;_l
&IL*’ =u;__~ +k l?*,(m-l) [
+Ft2)
(
&‘+l),
>I+i+’
+ F(l)
(q * W)im-”
(m,n=l,2
(q*~)(m-‘) n
(
E;~-“,
11 ,... ),
+ &“’
(q &$.,m-l)
>
>
(5.13)
c. v. PA0
872 where
I’*
=
l?(l)
+
and BLm)and c&~’ are the combined errors satisfying (5.10). Define
l-‘c2)
u; = I$) + aA2), p* = 1+ qp
M,* =
MC) + M(2) 7% 7 w; =
+ y*),
y*
=
p
+
p,
where crie) and iMAe’, C = 1,2, are the functions appearing in (4.11) and y([) 2 7::. have the following error estimate. 5.3. Let I$
THEOREM
be the solution
(5.14)
(y*+ 6%;*WQnll),
of (2.15) where F(un,vn)
Then we
is given by (4.5), and let
} be the computed sequences from (4.9) with W$,? = iin and VV,? = ii,. Assume {~~~‘},{Fv~? that Hypothesis (H) and condition (4.11) axe satisfied. Then there exist an integer m* and a norm in IV+’ such that
I/ II&Lu:,
I/
$$L,;
provided
+
II<-+ -
1 - lcw;
for all m 2 m*,
(n = 1,2,. . . , N),
(5.15)
that kwz < 1.
PROOF. Let E’“’
= Ktrn) -Ecrn) and Ecrn) = yim’ -I_&~). By subtracting the equations in (4.9) from the corresiondingnequatikrs in (cy3) and applying the mean-value theorem, we obtain
where (&, Tjn), etc. are some intermediate values. In view of the nonnegative property of A-‘,
(I’*
+ F$l’ + Fi2)), and Qn, the same reasoning as that in the proofs of Theorems 5.1 and 5.2 shows that
Addition of the above inequalities leads to the relation
IEkm)) 5 A-’ {k[(-I* + a;) I + M;Qn]
IE;m-l)l
+
Ie~m’l} ,
(5.16)
where Eim) = ELm’ + E,“), eP’ = Z&m’ + c&~) and y*, a:, and M,* are given by (5.14). Since (5.16) is of the same form as that in (5.6), we conclude from the definition of w;*Land the argument in the proof of Theorem 5.1 that relation (5.15) holds. I The error estimates given in Theorems 5.1-5.3 are based on “internal” perturbations (round-off error and the approximation of u:__~ by w,(m’)) and the initial perturbation eim’. These estimates yield the following numerical stability of the iterative schemes. THEOREM monotone
5.4. Let the conditions in Theorems 5.1-5.3 be satisfied for the corresponding function F(un, v,). Assume that there exists a constant 6 > 0 such that knw,
5 1- 6
and
knwz I 1 - S,
for all n = 1,2, . . . ,
quasi-
(5.17)
NumericalMethods
873
respectively. Then the corresponding iterative schemes (3.8), (4.2), and (4.9) are numerically stable with respect to combined internal and initial perturbations. PROOF. Under condition (5.17), estimates (5.5), (5.11), and (5.15) in Theorems 5.1-5.3, respec-
tively, imply that the error (Iji$“) - uzll + I&*’ -
It is easy to see from (5.4) that the requirement kw, < 1 in Theorems 5.1 and 5.2
is equivalent to k(o;, +
MdlQ,ll - P>I 1.
Since /_L2 0, the above relation follows from condition (3.15) if IIQnII5 1. Similarly, the requirement kwg < 1 in Theorem 5.3 follows from condition (4.11) if IIQnll < 1.
6. APPLICATIONS It is seen from Theorems 3.1, 4.1, and 4.2 that the main conditions for the monotone convergence of the maximal and minimal sequences {%@)}, {&m’} are the quasi-monotone property of the nonlinear function F(h,vn) and the existence of a pair of the corresponding upper and lower solutions. In the computation of these sequences, it is also necessary to choose l?, and k, so that condition (3.15) or condition (4.11) are satisfied. In this section, we consider three model problems (with linear boundary conditions) and give some basic techniques for the construction of upper and lower solutions and the choice of I’, and k,. These model problems demonstrate each of the three types of reaction functions treated in Sections 3 and 4. PROBLEM (a). The first model problem involves a Michael-Menton type of nonlinear function arising from enzyme-substrate reactions. This problem is given in the form (cf. [1,21])
ut - Du,, = au +
u(0, t) -
a(O)u,(O, t) =
‘4q* 4 1+q*u
g(O)(t)
‘ll(G0)
S(Ga
+
u(L, t) +
1ti(z),
(0 <
cA,(L, (0 <
2 <
L,
t) = g(l)(t)
t >
9
O),
(t > 01,
(6.1)
L),
2 <
where a zz a(t) is an arbitrary continuous function of t, and b E b(t) and ~(2, t) are continuous nonnegative functions of t and (z, t), respectively. For physical reasons, we assume that g(O)(t), g(l)(t), and $(cc) are continuous nonnegative functions in their respective domains. By the finite-difference approximations obtained in Section 2, a finitedifference system of (6.1) is given in the compact form (2.15) where the coefficient matrix A, possesses the property in Hypothesis (H)-(i). The function F(un,vn) is given by (2.12) with
Fi(ui,,, 2liJ = anu+ + *
z,n
+ Si,n + B&n,
(i=O,l,.*.,
p),
(6.2)
where & ii,*
= 0,
ifi=l,...,
=
p-l,
if i = p, and (n=
(6.3)
1,2,...).
Clearly, F(u,,v~) is a @-function for all u, 1 0, v n 1 0, and is nondecreasing in v,. This implies that the requirements of upper and lower solutions are given by Definition 3.1. It is obvious from the nonnegative property of ~i,~, _1cli,and gc’ (! = 0,l) that Q, = 0 is a lower solution. Let b, = (bn,. . . , b,JT,
ST3= (Sop, *. * 7Sp,n )+T
&I = (iO,n, 0,. . . , 0, &l)T,
c. v.
a74
and let U,
be the positive
solution
PA0
of the linear problem
(I + T-,&&J, = C-1 + kn[anUn + (bn + sn + in)], By Hypothesis
(H)-(i),
the existence
of a positive
solution
u,=ylJ.
to (6.4) is ensured
(6.4)
if k, a, < 1. It is
easy to verify from wi+/( 1+ vi,,) I 1 for wi+ _> 0 that fii, = U, is an upper solution. that the pair tin = U, and ii,, = 0 are ordered upper and lower solutions.
This shows
Since by (6.2),
the functions in (3.4) and (3.14) may be taken as yi+ = 0, CJ, = a,, that rn is the zero matrix and condition (3.15) becomes
Man + W < 1,
and M, = b,. This implies
(n = 1,2,...).
(6.5)
As a consequence of Theorem 3.1, we have the following conclusion: under condition (6.5) the model problem (6.1) has a unique finite-difference solution u; 2 0. Moreover, the sequences from (3.8) with -iI) = U,, 3:’ = 0, and I?, = 0 converge monotonrelation (3.16) with 11, = U, and ti, = 0, where U, is the positive
{@)}, {I_&~“‘}obtained ically to E: and satisfy solution
of (6.4).
PROBLEM (b) . As a second example, the form (cf. (1,6]) ut - Duzr =
we consider
a modified
u(a - bq * u) + ~(5, t),
diffusive
(0 <
5
logistic
equation
given in
L, t > O),
<
(6.6)
where the functions a, b, and s and the boundary and initial conditions are the same as those in (6.1). In this model, the function F(un,vn) is nonincreasing in v, for all (u,,v~) 2 (O,O), and its components are given by
Fi(ui,n, WUi,n) = Ui,n(G
-
(i = O,l,. . .,p).
bn~,n) + si,n + h,n,
(6.7)
In view of Definition 4.1 and Fi(O, v,) = .si,* + &,,, > 0 for any vn, the pair iin, ii, with ti, = 0 are upper and lower solutions if fin 2 0 and
(I + r,A,)iii,
2 tin-1 + k,F(k,
By (6.7), it suffices to choose tin = U,, lem (6.4) with b, = 0. Moreover, from
where U,
z(Ui,n,wi.n) =an - bnwi,n, the functions
0),
is the positive
z
(Ui,nr
solution
of the linear
prob-
wi,n)= -bnui,n,
in (3.4) and (3.14) are given by TOI
=x
=
max{O,
b, (q*q,
-a,},
where u’, is the maximum value of the components and A4,, condition (3.15) is reduced to
k, (an+ bnrn) < 1,
it& = b,u,,
an = a,, Vi,, of U,.
With
(n = 1,2,. . . ).
(6.8)
the above choice of on
(6.9)
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875
By an application of Theorem 4.1, we have the following conclusion:
under condition
(6.9),
equation (6.6) (with the boundary and initial conditions in (6.1)) has a unique finite-difference solution ui 2 0. Moreover, the sequences {@‘)},
{ail”‘}
obtained from (4.2) with $1
tin(‘) = 0, and I+, = ~~1 converge monotonically to ut and satisfy relation (3.16) the positive solution of (6.4) with b = 0. PROBLEM (c).
= U,,
where U,
is
When the modified reaction function in the diffusive logistic equation is replaced
by (g * u) (o - bzl), equation (6.6) becomes Ut
-
Du,,
(0 < x < L, t > O),
= (g * u) (a - bu) + s(x, t),
(6.10)
where a, b, and g are the same as that in (6.1). In this case, the components of F(un,vn)
are
given by (6.11)
Fi(ui,n, vi+) = u~i,n(on- b, ~i,n) + si,n + Gi,n.
Although the above functions are nondecreasing in ui,,, if ?.+ I an/b,,, this latter requirement imposes a very restrictive condition for the construction of a positive upper solution. To overcome this, we define
F!‘)(u. r,nr *
v.*,n
) =u n 21.r,n
and write F(un, v,) = F(‘)( u,,v,)
+sj,+ij.
*,n
7
+ F(‘)(u,,v~),
1
$d2’(u. r,n, z
v.r,n ) = -b n u. *,n v:l,lz
where F@(h,
v,),
(6.12)
e = 1,2, are the vectors
,r,n, vi+). Clearly, F(‘)(u,,v,) is nondecreasing in v, and Ft2)(u,, v,) with components F!“(u. * 2 (0,O). It is easy to verify from Definition 4.2 and (6.12) is nonincreasing in vn for (h,v,J that the pair tin = U, and ti, = 0 are coupled upper and lower solutions if U, is nonnegative and satisfies the relation
uoL +.
(I+ ~nAn)Un 2 Q-1 + k&n(q * U>, + sn + i&t], Since (q * U),
= Q,U,,
it suffices to choose U, as the positive solution of the linear problem
(I+ r,zAn - kn:nanQn)Un = Un-1 + k&n + &a),
(n = 1,2,. . . ),
IJo=+.
(6.13)
To show that a unique positive solution to (6.13) exists, we observe from Hypothesis (H), (2.2), and (2.16), that the elements ci;) of the matrix C,, G I + rnA, - knu,Q,
possess the properties
c!“.) %l -< 0 for i # j and (n) _ cp qj - 1 + Tn(f-Q,n- bi,, -
b&J - ha,,
j=o
ePjqi,j,n 2 1 - &a,,
j=O
for all i = O,l,. . . , p. This implies that if k,a,, < 1, then C,, is a diagonally dominant matrix and its inverse C;l exists and is a positive matrix (cf. [17]). Hence, a unique positive solution U, to (6.13) exists. Moreover, by the nonnegative property of Qn, this solution can be computed from the iteration process (I + ~-nAn)tJ~~~ = U;_,
+ (k~a,Q,z)U~?
+ k&n
+ &n),
(n = 1,2,. . . ),
(6.14)
uo=e, by the Thomas algorithm, where Uz_i
is the solution of (6.13) at the previous time-step.
In-
deed, since UT, = 0 is a lower solution of (6.13), the well-known monotone convergence theorem
c. v.
876
PA0
(or Theorem 3.1) ensures that the sequence {Ukm”} given by (6.14) with U?’ monotonically from below to a unique positive solution U, (cf. [11,19]). Since
dF!2) $+.n,2)i,n)
= -&%,n,
= 0 converges
1+2) *(%,nr
%,n)
= -bn%,n,
we see from (3.4) and (3.14) that
rt’,‘n= 0, y!2) = bn(q * U)i,n9 2,n where Dn is the maximum of the components Vi,,, of U,. satisfied if (n= k, (a, + b,T’j,) < 1,
A/l(l) =
a,,
Mi2’ =
b,ui,,
(6.15)
This implies that condition (4.11) is 1,2,...).
(6.16)
By Theorem 4.2, we have the following conclusion: under condition (6.16), equation (6.10) (with the boundary and initial conditions in (6.1)) h as a unique finite-difference solution UC 2 0. Moreover, the sequences {Ehm)}, {this’}
obtained from (4.9) with Xii’) = U,,
= 0, I’?’ = diag(#i,. . . , -ygi) converge monotonically where U, is the positive solution of (6.13).
&’
= 0, I$”
to UC and satisfy relation (3.16),
The construction of upper and lower solutions for the above model pro.blems can be used to compute numerical solutions whenever the physical parameters and the boundary-initial functions are given. Some of the numerical results for the special case q*u = u in the above model problems, including some test problems with known analytical solutions, can be found in [14].
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