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Error estimates for truncated domain approximations of travelling wave solutions of a combustion model
Nonlrnear Ana/ysrs, Theory, Methods Printed in Great Britain.
Vol. 17, No. 12, pp. ,177-1200,
& Appl~ot,ons.
ERROR ESTIMATES FOR TRUNCATED OF TRAVELLING WAVE SOLUTIONS
0362-546X/91 $3.00+ .OO “G 1991 Pergamon Press plc
,991.
DOMAIN APPROXIMATIONS OF A COMBUSTION MODEL
R. FERRETTI Dipartimento di Matematica, Universita di Roma “La Sapienza”, 00185 Rome, Italy
and B. LARROUTUROU Cermics, INRIA, Sophia-Antipolis, (Received 20 September
06560 Valbonne, France
1990; received for publication
13 March 1991)
Key words and phrases: Travelling waves, error estimates, combustion models.
1. INTRODUCTION THE MOST fundamental
problem in combustion theory concerns a steady planar pre-mixed flame in an infinite tube. In the simplest case where one assumes a one-step chemical mechanism and a unit Lewis number and when one uses the classical isobaric approximation, the steady (or travelling-front) solution of the reacting flow equations is described by the following simple model (see e.g. [3, 6, 7, lo]): in IR,
--uM + CU’ = g(u) i U(---co) = 0,
u(+w)
= 1.
(1.1)
Here, the C?’ function u is the normalized temperature defined on all of IR; 0 (resp. 1) is the normalized temperature of the fresh mixture (resp. of the burnt gases) at the left (resp. right) end of the infinite tube. The unknowns in (1.1) are both the temperature field U(X) and the positive scalar c, which measures the normalized gaseous mass flux across the steady flame: in some sense, c plays the role of an eigenvalue in this problem. In the differential equation (1.1) the terms u”, CU’ and g(u) represent respectively heat conduction, heat convection by the flow, and heat source released by the chemical reaction. We will assume that the function g has an ignition temperature 8 [see (2.11) below]. For numerical purposes, one has to approximate the solution (u, c) of (1.1): one considers a bounded domain [-a, a] (with a “large enough”) and solves for a truncated approximation of (1 .l). Several such bounded-domain problems are considered by different authors (see, e.g. [8, 9, 12]), all being of the following form: one seeks for u, E (?‘(-a, a) and c, E IF?satisfying: -u; subject
to one of the following
to one of the following
+ c, U:, = g(u,)
boundary
conditions
-U:, (-a)
+ c, u, (-a)
boundary
conditions &(a)
= 0
in (-a,
a),
(1.2)
at x = -a:
= 0
or 2.4,(-a) = 0,
(1.3)
at x = a: or u,(a) 1177
= 1,
(1.4)
1178
and to the additional
R. FERRETTIand B. LARROUTUROU
condition: u,(O) = 8.
(1.5)
Condition (1.5) is added in order to keep c, unknown in (1.2)-(1.5) (see the discussions of this point in [l, 51). Some of the problems (1.2)-(1.5) [depending on the boundary conditions (1.3) and (1.4) which are actually used] have been studied from a mathematical point of view. One can then show that the solution (u,, c,) of (1.2)-(1.5) converges as a tends to +w to a solution (u, c) of (1.1) which in addition satisfies u(O) = 8. In fact, investigating a bounded-domain problem of type (1.2)-(1.5) and studying the limiting behaviour as a tends to +co is a good way of proving the existence of a solution to the original problem (1.1) (see [2-5, 111). Because the existing convergence proofs rely on compactness arguments, they do not provide any estimate of the errors Ic, - cl and IIu, - ~11. It is precisely our purpose in this work to derive such error estimates, whose interest for numerical calculations is clear. Under the appropriate conditions on the nonlinear function g, we will prove that the convergence of there exist a positive constant K and an explicitly known positive (u, 9c,) to (u, c) is exponential: constant k such that, for a large enough, the following holds: Ic, - cl I KePka,
IIu, - ~ll~z~-~,~)I Keek”.
(1.6)
The precise hypotheses and results are stated in theorems 3.1, 3.2, 3.6 and 4.2 below. The paper is organized as follows. We first recall, in Section 2, some known results about problem (1.1). The new mathematical results, and in particular error estimates of the form (1.6) are presented in Sections 3 and 4. Our main result is presented in Section 4, where we show that using a new boundary condition at x = a [instead of one of the conditions (1.4)] accelerates the convergence of the corresponding truncated solution to (u, c) (see theorem 4.2 below). Lastly, we illustrate the various convergence (or divergence) results by showing some numerical examples in Section 5. 2. SOME
We recall in this section
several known
KNOWN
results
RESULTS
about
2. I. A comparison principle We begin with a comparison principle for solutions with (1.1). This result was proved in [2, 31.
problem
(1.1) and related
of initial value problems
questions.
(IVP) associated
PROPOSITION 2.1. Let g, and g, be two Lipschitz-continuous functions, and let x0, 6, cl, c2, 01~,CY~ be given real numbers. For i = 1,2, consider the solutions ut and u2 of the forward IVP:
-Ui’
I
+ CjUi' = gi(Uj)
ui(xlJ = 6,
If:
g1
5
on [x0, +w),
&(x(J) = a;. g2
on
14
+a),
Cl zc21
011 2 cY2> 0, then ut (x) r u,(x)
for any x > x,, such that u; > 0 and ui > 0 on [x0, x).
(2.1)
(2.2)
(2.3) (2.4)
Error estimates
for travelling
1179
wave solutions
Remark 2.1. The proof of this result, which we do not present in detail here, relies on inverting the functions ui and u2 in the intervals where they are monotone increasing. Setting x;(s) = u;l(s) (i.e. i(;(xi(s)) = s) for i = 1, 2 and for s in some interval [a, S’), we can define Zi(S) = (d/ds)xi(s) and vi(s) = Zi(S)~l = u,!(xi(s)). Then, the functions yi satisfy:
(2.5) and one can show that y, 2 y, in [a, S’], whence [2, 31 for the details.
zi I z2, x, 5 x2 and u1 1 u2. We refer to
Remark 2.2. With the assumptions of proposition 2.1, one can also show that, if one of the two inequalities (2.3) or (2.4) is strict, then U,(X) > uZ(x) for any x > x0 such that u; > 0 and u; > 0 on [x0, x). We can also state a similar
result for the backward
IVP.
PROPOSITION 2.2. Let g, and g, be two Lipschitz-continuous functions, and let x0, 6, cl, c2, a,, 01~ be given real numbers. For i = 1,2, consider the solutions ui and u2 of the backward IVP: -Ul’ + ciu;
= g,(u,)
on (-~,xJ,
(2.6)
u; (x0) = a,.
i U;(%) = 6, If: g, 5
on
g2
cl 2
t-m,
(2.7)
61,
c2,
(2.8)
(2.9)
o
then U, (x) L u2 (x) for any x < x0 such that u; > 0 and U; > 0 on (x, x0]. 2.2. The basic results for problem (1.1) Let us now recall the main existence and uniqueness in the whole article that the function g satisfies:
result for problem
g is Lipschitz-continuous 3 0 E (0, l),
g = 0 on (-03,191,
in R,
g’(1) < 0. are classical
in combustion
(2.11) (2.12)
in (2.1 l), 19is an ignition temperature. Moreover, we will always assume, below, that g is differentiable at the point s = 1 and satisfies:
(2.10)-(2.13)
(2.10)
g > 0 on (0, l),
g = 0 on [l, fco);
All assumptions
(1.1). We will assume
except for theorem
2.3
(2.13) theory.
R. FERRETTI and B. LARROUTUROU
1180
Remark 2.3. All results presented g is allowed with:
to be discontinuous
below could be straightforwardly extended to the case where at 8, i.e. to the case where g is Lipschitz-continuous on (0, 11, (2.14)
lim g(s) > 0, Sh0 and satisfies
(2.1 l)-(2.12).
The following
result has been proved
THEOREM 2.3. There exists a unique
by Berestycki
solution
et al. [5].
(u, c) of:
-UI’ + CU’ = g(u)
in IR, (2.15)
i u(-co) and this solution
satisfies
= 0,
U(0) = e,
u(+oo)
= 1,
c > 0 and U’ > 0 in I?.
We omit the proof, which relies on a shooting (see [2, 3, 51). Let us simply add that the solution U(X) = t!Iecx
method and uses the comparison (u, c) of (2.15) satisfies:
for all x 5 0,
principle
(2.16)
whence: U’(0) = 09. We also recall the following
result about
the asymptotic
(2.17) behaviour
of U(X) as x tends to +a.
PROPOSITION 2.4. Assume that the hypothesis (2.13) holds. Then there exists a positive k such that: u’(x) = kremrx + o(emrx), U(X) = 1 - kemrx + o(ePrx),
constant (2.18)
as x tends to +co, with:
r=
Jc”
- 4g’(l)
- c
2
(2.19)
> 0.
The proof of proposition 2.4 relies on the investigation of the linearized --u” + CU’ = g’(l)(u - 1) and on some classical results of ordinary differential theory (see [2, 31). Throughout the paper we will always call (u, c) the unique solution of (2.15). 2.3. A related initial value problem We will consider several times in the sequel the following with (2.15). For any y > 0 we call U, the solution of:
L
2.l; +
yu; = g(u,)
u,(O) = 8,
u{(O) =
We already know from (2.17) that U, = u on IR,. follows from Then, the following result essentially to [2, 3, 51 for the proof).
initial
value problem
on R+,
associated
(2.20)
ye. the comparison
equation equations
principle
(we refer
1181
Error estimates for travelling wave solutions
PROPOSITION 2.5. If y > c, then u_l > 0 on R, and u,(+m) = +a~. In particular, there exists a unique positive real number x,, such that u.,(x,,) = 1. If 0 < y < c, then u,(+-co) = --co and there exists a unique positive real number .C, such that uC(Q = 0, with u,(&) E (f9, 1).
We will use below the classical fact that, for any x > 0, u,(x) and UC(X) continuously on y. We will also need the following property (see [2, 31). LEMMA
2.6. The solution
uy of (2.20) satisfies
u_l 5 y on the interval
lx E R, , u,(x)
depend
I
11.
2.4. Truncated solutions Finally, we end this section by considering some truncated solutions of (2.19, i.e. solutions of a problem which is similar to (2.15) but posed on a bounded domain (-a, a). Such solutions have been considered by several authors, either from the mathematical point of view or for the numerical approximation of problem (1.1) (see, e.g. [3, 5, 8, 121). In particular, two bounded domain approximations have mainly been considered in the literature. The first one consists of finding (ii,, co) satisfying: - 22; + i +:(-a) while the solution
in (-a, a),
coii,: = g(z7,) + C*,ii,(-a)
(u,, c,) of the second
problem
-24; +
c,u:, = g(u,)
u,(-a)
= 0,
L
a*(O) = 0,
= 0,
(2.21)
ii,(a) = 1,
satisfies: in (-a, a),
u,(O) = 8,
(2.22)
u,(a) = 1.
We emphasize that any confusion between the notation u, defined by (2.22) and the notation 11~defined by the IVP (2.20) should be avoided in the sequel. The following existence, uniqueness and convergence results are proved in [5] for problem (2.21) and in [2, 31 for problem (2.22). THEOREM 2.7. For any a > 0, there exists a unique solution (ii,, co,) of (2.21); satisfies ii: > 0 in (-a, a). When a tends to +m, ti, converges to u in C$,,,(lR) and C~ converges to c. THEOREM 2.8.
There exists Li:> 0 such that problem (2.22) has a unique only if a > CT;this solution satisfies U; > 0 in (-a, a). When a tends to fcu, u, converges to u in C&(R) and c, converges 3. EXPONENTIAL
ERROR
solution
this solution
(u,, co) if and
to c.
ESTIMATES
In this section, we will derive error estimates for three different truncated the solutions (ii,, c,) and (u,, c,) considered in Section 2.4 above.
solutions,
including
3.1, Zero-flux condition at the left boundary We begin with the simplest case, where a zero-flux condition is imposed at the left boundary x = -a, i.e. we consider the solution (ii,, CJ of (2.21). We will prove the following result.
R. FERRETTI and B. LARROUTUROU
1182
THEOREM 3.1. Assume that g satisfies (2.13) and is monotone decreasing in some left neighbourhood of 1. Then, there exist two positive constants a,, and K such that, for any a > a,, the solution (ii,, CJ of (2.21) satisfies the following estimates:
ICa- cl
I
Ke-““,
IIii, - uII~~~-~,~)5 Ke-‘“,
(3.1)
where r is given by (2.19).
Remark 3.1. The fact that g is decreasing
in some neighbourhood of 1 follows from assumption but does not follow from (2.10) to (2.13).
(2.13) if g is C?’ in this neighbourhood,
Proof. We divide the proof into four steps. Step 1. An upper bound
for C~ - c. It is easy to see that U,(X) = Be’“* for proposition 2.5, we see that (5, > c, that determined by the equation Q = a (we comparison principle then shows that ii, 2 (because x,, decreases as a function of y). Thus, we may invert ii, and define x,(s) z,(s) = x:(s), y,(s) = l/(z,(s)) = iiL(x,(s)). y(s) = l/(z(s)) = u’(x(s)). Then:
all x E [-a, 01, whence ii,: (0) = C~8. Then, from increasing, and that c0 is ii, = u,~ is monotone use here the notations of proposition 2.5). The u in [0, a], and that ?a decreases as a function of a by the relation Z&(X,(S)) = s, for s E [8, 11, and set We also define x(s) by u(x(.s)) = s, and z(s) = x’(s),
y:,(s) =
g(s) r, - Y, (s)’
(3.2)
Y,(B)
Ge,
(3.3)
= %(a).
(3.4)
also shows (see remarks
2.1 and 2.2) that ya (s) > y(s) for
y’(s) = c - ;+, Y(B) =
co,
Y(l) = 0, Since ?a > c, the comparison principle all s E [Q, 11; therefore, we have:
Y,(l)
=
(3.5) Integrating
this relation
over the interval y,(l)
[0, l] yields:
- ~(1) 2 (co - c)(l - 0) + Y,(O) - Y(O),
(3.6)
or equivalently: - -ciiiL(a). C,
(3.7)
Step 2. Exponential
estimate for i;, - c. Since ?a decreases as a function of a, there c0 I C?< +os for all a large enough. Now consider IVP:
-w; +
Ew:, = 0
exists an upper bound for ~~a: we have the solution w, of the following backward in (-co, a), (3.8)
i W,(Q) = 1,
WA(U) = &(a).
Error
The comparison Then, we get:
principle
estimates
for travelling
for backward
2.2) shows that w, 2 ii, on [0, a].
IVPs (proposition
a,:(4
u(x) 5 U,(X) 5 w,(x)
1183
wave solutions
= 1 + ~
?
(e
p(wpa)
- l),
for all x E [0, a]. On the other hand, we know from (2.18) that we can find a positive K such that: 0 5 1 - U(X) I Ke-‘” for all x 2 0. Writing
(3.9) for x = a - 1 and using (3.10),
constant (3.10)
we can write:
G(Q) -c^ - 1) 2 u(a - 1) L 1 - Ke-r(a-l), (e c
l+p
(3.9)
(3.11)
and an estimate of the form c0 - c 5 Ke-‘” follows from (3.7) (here and in the sequel, the notation K denotes a positive real number which does not depend on a, but which may not be the same at each time it appears). Step 3. Exponential estimate for ii, - U. Let us deduce from the previous step an estimate on ii, - U. We already have from step 1 an estimate on y, - y: since y, - y is increasing on the interval [f?, 11, we have: 0 I y,(s)
- y(s) I y,(l)
5 Ke-‘“,
- y(1) = a:(a)
(3.12)
for all s E [e, 11. Starting from this inequality, we are going to successively derive estimates on z - z,, x - x, and lastly on ii, - U. Let s0 E [0, l] be a fixed number, to be adequately chosen later. For s E [e, s,], we have:
0 5 44 - z,@) =
Y,(S) - Y(S) I y(s)y
(s)
a
Y,M
- Y(S)
min y(Q2’ B
(3.13)
so that there exists K > 0 such that: 0 I
Integrating
this inequality
z(s) - z,(s) 5 Ke-'"
for all s E [0, so].
(3.14)
for all s E [0, s,].
(3.15)
on (0, s), we get in turn:
0 I
x(s) - x,(s)
Beside this, we know from lemma
I
Ke-'"
2.6 that ii; I Ca I E. Then,
0 I &(X(S)) - U(X(S)) = ii, (x(s)) - i&(x,(s)) from which we deduce
for any s E [e, s,], we have:
I @x(s) - x,(s)),
(3.16)
that: 0 I ii,(x)
- U(X) % Ke-‘”
v x E 10, X@,>l~
(3.17)
Now, it remains to show that the same inequality holds for all x E [x(+,), a]. But we already know from (3.10) that ii,(a) - u(a) I Ke-‘“. Let us therefore examine whether the difference ii, - u can have a local maximum at a point xi in the open interval (x&J, a). If this is the case, we have: ii; I U”(X,), (3.18) ii,’ = u’(x,), 0, (x1) > U&I) > so 7
R. FERRETTI and B. LARROLJTUROU
1184
which implies g(ii,(x,)) 2 g(u(xi)). that g is decreasing on the interval
If, from the outstart, we have chosen s, large enough [s,, , 11, we get a contradiction. Therefore, we obtain:
0 I ii, - u 5 Ke-‘” The same property known.
obviously
Step 4. Exponential To end the proof, differential equations
holds on the interval
on [0, a]. [-a,
so
(3.19)
0] where both u and ii, are explicitly
estimate for ti: - u’. it remains to show that the estimate (3.1) holds in C2 norm. From satisfied by u and ii,, we see that we just have to check that:
the
la;(x) - u’(x)1 I Ke-‘” (3.20) for all x E [-a, a]. Firstly, checking (3.20) for x E [-a, 0] is obvious. Also, arguing as we did above to deduce (3.17) from (3.1.5), we see from (3.15) that (3.20) holds for allx E [0, x(s,)]. Beside this, we have: Iii:(a)
- u’(a)\ I Ke-‘”
(3.21)
from step 2 above and (2.18). Thus, it remains to examine the case where the difference ii; - U’ is extremal at some point xi in the open interval (x&J, a). In such a case, we have u”(x,) = ii:( and we get: ]COi$(Xi) - CU’(Xi)I = g@(Q)
- &%(x1))
from (3.19) and because g is Lipschitz-continuous. But it is easy to deduce (ii; - u’(x,)l I Ke-‘“, which concludes the proof. n 3.2. Dirichfet condition at the left boundary Let us now come to the solution (u, , c,) of problem
(3.22)
5 Ke-‘“,
from (3.22) that
(2.22). We will prove the following
result.
THEOREM 3.2. Assume that g satisfies (2.13) and is e’ in some left neighbourhood of 1. Then, there exist three positive constants a,, K and r’ such that, for any a > a,, the solution (u, , c,) of (2.22) satisfies the following estimates:
Ic, - cl 5 Ke-““,
(Iu, - z.JII~+~,~)s Ke-““.
(3.23)
Remark 3.2. Carefully
examining the proof of theorem 3.2 below, one can see that the constant r’ in (3.23) is smaller than r given by (2.19), and that for any E > 0, r’ can be chosen in the interval (i - E, ?) where F = min(r, c), provided that a, is taken large enough. Therefore, the “asymptotic rate of exponential convergence” F of the truncated solution (u, , c,) is explicitly known, and may be smaller than the rate of exponential convergence r of the truncated solution (ii,, Q) of (2.21).
Proof. We divide it into two steps. Step 1. Exponential estimate for c, - c. Let a be large enough so that problem (2.22) has a solution
(u, , c,). For x < 0, u,(x) is given by:
e=,x _ e-V U,(X) = e
’
(3.24)
> c,e.
(3.25)
1 _ e-c,a
and we have:
Error
estimates
for travelling
Let (ii,, CJ be the solution of (2.21). If we suppose comparison principle (proposition 2.1 and remark Therefore, we know that c, I co. On the other hand, if we assume that:
1185
wave solutions
that c, 2 c~,, we get u:(O) > ii:(O), and the 2.2) yields u,(a) > ii,(a), a contradiction.
(3.26) then u:(O) 5 u’(0) and c, < c. The comparison principle then another contradiction. Thus, (3.26) does not hold and we obtain:
implies
that
u,(a)
c(1 - e-caa) < c, 5 Co.
< u(a),
(3.27)
Since theorem 2.8 shows the existence of a lower bound on c, (c, > c0 > 0), and using theorem 3.1, we obtain: c - ce-‘@ I c, 5 c + Ke-‘“, (3.28) which proves the desired
estimate
for Ic, - cl.
Step 2. Exponential estimate for u, - U. If c, L c, then the exponential estimate for u, - u can be derived using exactly arguments as in step 3 of the proof of theorem 3.1. Thus, it remains to prove a similar in the case where c, < c. Assume therefore that c, < c. Define the functions x, and y, by u,(x,(s)) u; (x0 (s)) = ya (s) for all s E [0, 11. We will also use the functions x and y defined in of theorem 3.1. We already know from (3.27) that: 4 (0) = Y,(B) > Y(B) = u’(O),
the same estimate = s and the proof
(3.29)
and that: Y,(l) We claim that these inequalities
(3.30)
> Y(1) = 0.
imply that: on [0, 11.
Y, > Y
(3.31)
Indeed, if y,(s) = y(s) for some s E (0, l), then the difference y;(s) - y’(s) is negative (from the differential equations satisfied by ya and y), and the inequalities (3.29)-(3.30) cannot hold together (notice that the comparison principle cannot be directly used since c, < c). Now let s, E [0, l] be a fixed number, to be adequately chosen later. Since we know that: Y:, (4
- Y ‘(4 =
G -
c+
m
Y,(S) -
Y(4 Yn @lY@) ’
(3.32)
we obtain: Y;(s) - Y’(S) 5 R(Y,@) - Y(S)) for all s E [e, s,], with R=
max g(s) e
~(4~ ’ 0
(3.33)
R. FERRETTIand B. LARROUTUROU
1186
Using Gronwall’s
lemma,
we get: r,(s)
.d@leR’“-o’
- Y(S) 5 [u, (0) -
(3.34)
for all s E [0, s,]. But it is easily seen from (3.25) (3.28) and (3.29) that y,(B) - y(B) 5 Ke-“” for some constant r’ independent of a, and we can write that: 0 < y, - y 5 Ke-“”
(3.35)
on [e, ~1.
Now, arguing as in the proof of the comparison principle (see remark 2. l), one can deduce from (3.31) that u, > u in (0, a). Then, using the same arguments as in step 3 of the proof of theorem 3.1, we can prove that: u,(x)
- u(x) I Ke-“”
for all x E [0, x&J].
(3.36)
Since 0 I u,(a) - u(a) 5 KC”“, it remains to examine the case where the difference u, - u has a local maximum at some point xi in the open interval (x(sO), a). If this happens, we have: u, (x1) > u(x,) > so 3
ul(xJ
U:,(Xl) = u’(x,),
5 U”(X,),
(3.37)
which yields: c,u’(x,) If, from the outstart,
- g(u,(x,))
5 cu’(x,)
we have chosen so close enough
(3.38)
- g(u(x1)). to 1 so that:
max g’(s) < 0 s,5s51 (which in particular
implies
that g is decreasing
0 I g(u(x,))
- g(u,(x,))
(3.39)
on the interval 5 (c - c,)u’(x,)
[so, 1]), we get: 5 Ke-“”
(3.40)
(we recall that u’ is uniformly bounded from lemma 2.6). Since the inverse of g is Lipschitzcontinuous in the interval under consideration from (3.39), we deduce from (3.40) that u,(xJ - u(xJ 5 Ke-““, whence: 0 I u,(x) The end of the proof
- u(x) 5 Ke-“”
is then analogous
to step 4 of the proof
3.3. Neumann condition at the right boundary We now wish to examine the following problem, atx = a: --a; + &,ti,: = g(ti,) in (-a, ( A,:(-a)
+ &z&(-a)
for all x E [0, a].
= 0,
of theorem
with a homogeneous
(3.41) 3.1.
n
Neumann
condition
a), C,(O) =
8,
z&(a) = 0.
(3.42)
We will see below that additional difficulties related to the behaviour of g in the neighbourhood of B occur for error estimates of the form (1.6) to hold for (ti,, &). We first need to investigate the existence and uniqueness of (ti,, 2J since no result like theorem 2.7 is known about this problem in the literature. It is easy to see that any solution (a,, &) of (3.42) satisfies: Z&(X) = e&X (3.43) for all x E [-a, 01, whence C;(O) = &0. Using the notations of proposition 2.5, we see that ti, = Uf,, and that solving problem (3.42) amounts to find & < c such that $, = a. This is why we need to prove some properties of the application 8: y - .?? defined in the interval (0, c). This is the object of the next two results.
Error estimates for travelling wave solutions PROPOSITION 3.3. The application
in the interval
8: y H k, (with L??defined
1187
in proposition
2.5) is continuous
(0, c) and satisfies: lim& Y/‘C
= +w.
(3.44)
Proof. Notice first that proposition 2.5 implies that, for all y E (0, c) and for all x > 0 with u;(x) has the sign of the difference ,?,, - x. xzg7,, Now, we argue by contradiction. Suppose that x is not continuous at some point y0 E (0, c). Assume that there exists a sequence (y,) with lim yn = y0 such that lim j$,, > &,. Then, n-+m n-+m for n large enough and for some x0 > &,, we have iYy, > x0. But we have $(x0) > 0 for all n large enough, whence u&(x,) 2 0 by the continuous dependence of uy with respect to y, and this contradicts the fact that x0 > &,. Arguing in the same way, we also get a contradiction if we assume that there exists a sequence (y,) with nkmm yn = y,, such that lim gYy,< iYo. Therefore, the application 2 is continuous. n++C= The argument for (3.44) is really the same. If there exists a sequence (y,) with y, < c and lim yn = c such that lim k,,” < +co, then we choose x0 > 0 so that gTy, < x,, for all n and we n-+m a-+m get ub,(xJ < 0 for all n, whence uL(xO) I 0, a contradiction. n PROPOSITION 3.4. Assume
then the application
that g has a right first derivative X: y ++ jZ7 satisfies:
g’(P)
at the point 8. If g’(P)
lirn,?? = ’ 2@@7’ YLO If g’(0’)
> 0,
(3.45)
= 0, then: lim$ YhO
= + co.
(3.46)
Proof. Notice
first that it follows from the comparison principle (proposition 2.1) that = max u,(x) is a monotone increasing function of y E (0, c) (see [2, 31 for the details). X>O Choosing a fixed y. E (0, c), we set so = ZL,,(&J E (0, 1). Consider first the case where g’(P) > 0. We can then find two positive constants m and M such that: m(s - e) < g(s) < M(S - e) (3.47) u~(,Q
for all s E (0, so]. Assume
that y < min(2&,
yo). Then,
consider
the solution
wy of:
e),
w:,(&)
which is given by: WY= e + (24,(2?) - e)exp(Y(x with a,”
= JM
2 9Y))(c0s]c0r(x
- i?)] - &sin[wr(x
- L,)]),
(3.49)
- y2/4. Setting: _Y,=Z?7+$[arctan(3
we see that w,(y,)
(3.48) = L$(.q) = 0,
- 7r],
= 8 and that wy > 8, w; > 0 in the interval
(y,, 2.J.
(3.50)
R. FERRETTIand B. LARROUTUROU
1188
Now, we claim that w, s uy on the interval [y,, &I. In fact, this does not directly follow from the backward comparison principle (proposition 2.2) since w~(Q = uk(.Q = 0. But we can notice that w, 5 uy in some left neighbourhood of & since w,“(&,) < us from (3.47) and our choice of y. Therefore, if there exists x = maxly E [y,, Q, w,(y) > u,(y)), we necessarily have w?(x) = u,,(x) and w;(x) 5 u;(x); we even have the strict inequality w;(x) < u;(x) since the identity w;(x) = u;(x) would imply w;(x) < u,“(x) and wr < uy in some left neighbourhood of x. Then, we can apply the forward comparison principle (proposition 2.1 and remark 2.2) to get a contradiction. Therefore, we have 8 = w,(y,) 5 u,(y,), which shows that yy 2 0, whence: (3.51) In a completely
similar way, for y < min(2&, -u,” +
yus,=
yO), we can consider the solution uy of:
m(u, - e),
(3.52) and define: .z,=a+$[arctan(F) with CIJ,”= w. that
-7r],
Then we can show that u, 5 u, on the interval
(3.53)
[z,, ZJ,
which implies (3.54)
Now, (3.54) shows that 2;;,is uniformly bounded as y tends to 0: we claim that this implies that: lim u,(&,) = 19. (3.55) Y’O Indeed, for any sequence (y,) with ,,li_mm yn = 0, we can extract a subsequence (still denoted (y,J) such that the sequence (&) converges to some positive limit x0. Then, we obtain lim uJ&,) = uo(xo) = 19since the functions u,, are equicontinuous from lemma 2.6, and n++CC (3.55) follows. We can now conclude the proof. Using (3.55) we see that, for any y small enough, we can choose two constants m7 and M, such that (3.47) holds for all s E (0, K,(Q], and so that: lim m, = ;y My = g’(0’). L YLO
(3.56)
Then, examining the limits as y tends to 0 of the inequalities: -&[ 71- arctan( % 7
ai,r&[n-arctan(22)],
which are the analogues of (3.51) and (3.54), we easily obtain (3.45). Lastly, in the case where g’(B+) = 0, we use the same construction
(3.57)
with lim M, = 0, and
(3.46) follows from examining the limit of the left inequality (3.57) as y ten&‘:0 0.
n
Error
estimates
for travelling
1189
wave solutions
Remark 3.3. As shown by the previous proof, the conclusions of proposition 3.4 could be reached from a more heuristical point of view by considering the linearization of the IVP (2.20) in the neighbourhood of the point (x = 0, uy = 0). Remark 3.4. discontinuous
we can
In connection with remark 2.3, at 0 and satisfies (2.14), we have lim& YLO
add
that,
in the
case
where
(3.58)
= 0.
To see this, it suffices to use the same construction lim m, = +oo; (3.58) then follows from examining YbO as y tends to 0.
g is
as in the proof of proposition 3.4, with the limit of the right inequality (3.57)
Remark 3.5. We already said that the application y E (0, c) - u,(&) is monotone increasing in monotone the interval (0, c). However, the application x: y - & itself is not necessarily increasing in (0, c) (see Section 5 below for an example where x is not monotone). Propositions
3.3 and 3.4 immediately
provide
an existence
COROLLARY3.5. Assume that g has a right first derivative Then, there exists a positive constant ci such that problem if and only if a L ri. Proof. As already
said, solving
problem
(3.42) amounts
&, = g(&) It is clear from the properties of the application only if a 2 ci = min 2(y) > 0. n YE[O,c)
result for pr’oblem (3.42).
g’(0’) at the point 8, with g’(B+) > 0. (3.42) has (at least) a solution (6,) to)
to find & < c such that
= a.
(3.59)
2 that the equation
(3.59) has a solution
if and
Remark 3.6. It is clear from the preceding proof and from the properties of _$?that proving a uniqueness result for problem (3.42) would be equivalent to proving the monotonicity of the application 2. Thus, the solution of (3.42) is not unique in general (see remark 3.5 above). We are now able to derive the following
error estimate
for problem
(3.42).
THEOREM 3.6. Assume that g satisfies (2.13) and is monotone decreasing in some left neighbourhood of 1. Assume moreover that g has a right first derivative g’(@) at the point 0, with g’(0’) > 0. Then, there exist three positive constants ao, K and r’ such that, for any a > a,, any solution (a,, 2,) of (3.42) satisfies the following estimates: Ii& - cl 5 Ke-““,
I/z?, - ul(,z(_,,,)
I Ke-““.
(3.60)
Remark 3.7. Carefully examining the proof of theorem 3.6 below, one can see that the constant r’ in (3.60) is smaller than r given by (2.19), and that, for any E > 0, r’ can be chosen in the interval (r - E, r) provided that a, is taken large enough. Therefore, the “asymptotic rate of exponential convergence” of the truncated solution (a, , 2J is the same as the rate of exponential convergence r of the truncated solution (ii,, Q of (2.21).
1190
R. FERRETTIand B. LARROUTUROU
Proof. In the sequel, (fi,, i$) denotes any solution of (3.42) for a large enough. It is already clear from (3.59) and proposition 3.3 that & converges to c (and therefore that li, converges to u in e,‘,,(R)) as a tends to +a~. We divide the proof into two main steps which are similar to steps 1 and 2 of the proof of theorem 3.1. Step 1. An upper bound
for c - F, . For s E [0, fi, (a)], we define x(s), z(s), and y(s) as in the proof of theorem 3.1, and we now set &(X,(S)) = s, z,(s) = x;(s) and y,(s) = l/z,(s) = &(x,(s)). Then it is easy to see that y, < y, whence y’ - y: 2 c - ea on the interval [B, &(a)]. Integrating this inequality and using the values of y(0) and y,(8), we obtain: (3.61)
Y(fi, (a)) - Y, (fi, (a)) 2 (c - G)&i (a).
Now, as a simple consequence of De L’HBpital’s theorem, we deduce from the equation (3.2) satisfied by y that y is differentiable at the point 1, with y’(1) = -r, where r is given in (2.19). Hence there exists a positive constant K such that: y(s) 5 K(l Then,
- S)
(3.62)
for all s E [e, 11.
using the facts that z&(a) > 0 and y,(ti,(a)) c - F, 5 $(l
= 0, we obtain
from (3.61): (3.63)
- G,(a)).
Step 2. Exponential
estimate for c - &. Since & converges to c as a tends to +co, we can find a positive lower bound for &: there exists c” > 0 such that F, > ? for all a large enough. Then, choosing a fixed real number x0 > 0 and setting +(x0) = sO, we have: (3.64) ii, 2 so > 8, for all a large enough.
Beside this, there exists a positive constant
m such that, for all s E [so, 11: (3.65)
g(s) 2 m(1 - s). Consider
the solution
w, of the following
backward
IVP:
- wI( + cw; = m(1 - wU)
in (-03, a), (3.66)
w, (a) = & (a) >
w;(a) = 0,
which is given by: w,(x)
with:
r, =
=
1
_
(‘i ~~~)(r2er1+” - rler2cx-a)),
c--<.
r 2
2
=c+dzzi>o 2
(3.67)
(3.68)
Then it is easy to check that w: > 0 on the interval (0, a), and we can argue as in the proof of proposition 3.4 to show that li, 5 w, on [0, a]. From (3.64), this implies that w,(x,) > sO, which writes: (1 - s&2 - rl) (3.69) 1 - ii,(a) 5 r2erl(~O-a) _ rl e~2(w-~) ; using now (3.63), we obtain
the desired
estimate
for c - &.
Error
estimates
for travelling
1191
wave solutions
The end of the proof is now completely analogous to steps 3 and 4 of the proof of theorem 3.1, with the only difference that we now have CO< c, fi, < u, whereas we had C~ > c, ii, > u n in the context of theorem 3.1. We leave the details to the reader. Remark 3.8. If g’(Q’) = 0, it is clear from proposition 3.4 and corollary 3.5 that problem (3.42) has at least two solutions for all a large enough. One of these solutions converges to (u, c) and satisfies the error estimates (3.60), whereas one of these solutions has a completely different behaviour: as a tends to +co, C0 converges to 0 and fi, converges in C&(R) to the constant 8. 4. IMPROVED
We now consider solution of:
a fourth -u;
ERROR
bounded-domain
ESTIMATE
problem.
We will henceforth
call (u,, c,,) a
in (-a, a),
+ C,U,: = g(u,)
f -&(-a)
+ c,u,(-a)
= 0,
h(O) = 0,
2
i
u;(a) =
J
‘a - 4g2’(1) - ‘“(1
(4.1)
- u,(a)).
The mixed Dirichlet-Neumann condition at the right boundary x = a has been chosen in order to mimic the behaviour of the true solution u since we know from (2.18) that: Jc’ u’(x) lim x-+m 1 - u(x) = r =
- 4g’(l)
- c . ’
2
(4.2)
thus, the above mixed condition seems to be the best condition to impose at x = a in order to achieve the “transparency” of the boundary. We will see in fact that (u,, c,) converges faster to (u, c) than the other truncated solutions. 4.1. Existence Let us begin by proving the existence of a solution to (4.1). In the sequel, when problem (4.1) has a solution (u, , c,), we will set: r, = About
problem
Jc,z - 4g’(l)
- c, .
2
(4.1), we first prove the following
PROPOSITION 4.1. Assume that g has a right first g’(@) > 0. Then problem (4.1) has (at least) a solution B is defined in corollary 3.5. Moreover, for all n a> a&jzj’
(4.3)
result. derivative g’(@+) at the point 8, with (u,, c,) for all a L 6, where the constant
let (u,, c,) be any solution of (4. l), and call (ii,, ca) the solution of (2.21). Then there exists a solution (ti, , e,) of (3.42) such that the following inequality holds: F” < c, < Cg.
(4.4)
1192
R. FERRETTI and B. LARROUTUROU
Proof. Let a I 6, and call (fi,, &) a solution of (3.42), and (ii,, co) the solution of (2.21). On the interval (&, cJ, consider the application [the notation U, is still defined by (2.20)] CB:y-
u;(a) ( 1 - u,(a)
It is easy to see that the application
Jr” -
CJlis continuous
lim CR(y) < 0, yLPL1 Thus, there exists a real number c, (4.1) [here, the notation uC, is also Now, let (u,, c,) be any solution the mixed boundary condition and fore, we have ??(c,) > a. If
- 4g’(l) 2
- Y
(4.5)
>.
on (i$, co,) and satisfies:
lim CR(y) = +w. y/E,
(4.6)
E (2*, CT=,) such that aZ(c,) = 0, and (uC,, c,) is a solution of defined by (2.20)]. of (4.1); it is easy to see that u, = uC,, and to deduce from proposition 2.5 that c, < I?~, u,(a) < 1, u:(a) > 0. ThereTt “2@@5’
this shows from the properties of the application & E (0, c,) such that 2(&) = a, which concludes
8 (propositions the proof. n
Remark below).
for problem
4.1. We have no proof
remark
4.3
THEOREM 4.2. Assume that g satisfies (2.13) and is C?’ in some left neighbourhood of 1, and g”(1) exists. Assume moreover that g has a right first derivative g’(P) at the point 8, g’(P) > 0. Then, for any E > 0, there exists r’ E (r - E, r), where r is given by (2.19), and positive constants a, and K such that, for any a > (I~, any solution (u, , c,) of (4.1) satisfies following estimates:
that with two the
4.2. Error estimate The error estimate
of uniqueness
3.3 and 3.4) that there exists
for the solutions
(4.1) (see however
of (4.1) is given in the following
]c, - cl I Kem2”‘,
I/u, - u~~~+~,~~I Ke-2”a.
result.
(4.7)
Proof. Let E > 0. It follows from (4.4) and remark 3.7 that we can find r’ E (r - E, r) such that (notice that li, < U, < ii, in (0, a) from (4.4) and the comparison principle): ]c, - cl 5 Ke-““,
IIu, - uII~o~-~,~)5 Ke-““.
(4.8)
Therefore, our purpose is to prove that these estimates can be improved to yield (4.7). We are going to prove (4.7) in the case where c, > c. We leave it to the reader to check that the proof is identical in the case where c, < c (with the obvious modifications which come from the fact that U, - u is then negative). We will have two steps. Step 1. For s that y, y’(1) =
Exponential estimate for c, - c. E 10, U, (a)], we define as before x(s), z(s), and y(s) and x,(s), .zU(s), and ya (s). We know - y is positive and decreasing on the interval [6, u,(a)]. Moreover, we know that -r, and it follows from the existence of g”(1) and from De L’Hopital’s theorem that y
Error
estimates
is e2 in some left neighbourhood a positive constant K such that:
for travelling
of 1. Then,
1193
wave solutions
using the mixed boundary
condition,
we can find
0 5 lYa(4 64) - Y(%WI r(l - %7wl + (r - r,)(l - x7@))
5 IYe47(4) 5
K(1 - 2.4,(a))2 + K(c, - c)(l - u,(a)),
(4.9)
where r, < r is given by (4.3) (we have used the fact that the application y y -JY2 - 4&(l) 2 is decreasing
and Lipschitz-continuous).
Hence,
y,(u,(a))
Now, we have the analogue
- Y
using (4.8) and (2.18), we get: I Ke-2”a.
- y(u,(a))
(4.10)
of (3.5), that is: (4.11)
y;(S) - y’(s) 1 c, - c for all s E [e, u,(a)].
By integration
over this interval,
we obtain: (4.12)
(c, - c)u, (a) 5 Y, (u, (a)) - Y(%(@), which together
with (4.10) gives the desired
estimate
for c, - c.
Step 2. Exponential estimate for u, - u. Let se be chosen as in step 3 of theorem 3.1. Using the same arguments that: X.;;~~Ojl [u,(x)
- 441
as there, we can prove
5 Kee2”“,
(4.13)
and that u, - u cannot have a maximum in the open interval (x&J, a). Then examine the case where u, - u has a maximum at the point x = LI. If this happens, u;(a) 2 u’(a), which means that: Y, (u,(a))
it remains to then we have (4.14)
2 Y(U(Q)).
Since y is C?’ in the neighbourhood of 1 with y’(1) < 0, we can now choose E such that max y’(s) < 0. For a large enough, we have 1 - E I u(a) I u,(a) < 1, whence: s E [l--E, 11 Y(0)) Thus,
(4.14) and (4.15) show that y(u(a)) y(u(a))
Since the inverse
E [y(u,(a)),y,(u,(a))], - y(u,(a))
of y is Lipschitz-continuous u,(a)
(4.15)
2 Y(%(Q)). and (4.10) now implies:
I Ke-2r’a.
in the interval
(4.16) [l - E, 11, we obtain:
- u(a) 5 Kem2”‘,
(4.17)
which proves that: Kl - u 5 Ke-2r’a The end of the proof
is then similar
on [0, a].
to step 4 of the proof
of theorem
We end this section with the following additional result on the solution the sign of c, - c for the solution (u,, c,) of (4.1).
(4.18) 3.1.
H
of (4. l), which gives
R. FERRETTI and B. LARROUTUROU
1194 LEMMA 4.3.
large enough,
Assume that the hypotheses of theorem 4.2 hold and that g”(1) # 0. Then, the difference c, - c has the same sign as -g”(l).
Proof. A straightforward
calculation
and the use of De L’HBpital’s
y”( 1) = - $2
(4.19) for a large enough,
Y(% (a)) < r(l - u,(a)).
y ~
= r,(l
show that:
-#‘.
Assuming that g”(1) < 0, we therefore get y”(l) < 0. Then, to 1 and we can write (we recall that y’(l) = -r):
If c, I c, we havey(u,(a)) r _~,(u,(a)) this is impossible since the application
theorem
- u,(a)),
JYZ- W(l) -
for a
and we obtain
u,(a)
is close
(4.20)
r, < r from (4.20). But
Y
-I
is decreasing.
Remark 4.2. In the combustion increasing,
we have g”(1) = -f’(l)
context, where g is of the form g(s) = (1 - s)f(s) with f < 0. Thus, any solution (u,, c,) of (4.1) satisfies c, > c.
Remark 4.3. Let us also add that, if g’ is constant
in some left neighbourhood of 1, then for all a large enough (u, c) is the only solution of (4.1). Indeed, if g(s) = r(1 - s) in some interval [I - E, 11, there exists a constant k > 0 such that U(X) = 1 - keerx for all x large enough, so that (u, c) is a solution of (4.1) for a large enough. Moreover, we also have y(s) = r(1 - s) for all s E [l - E, 11, and we can argue as in the proof of lemma 4.3 to show that (4.1) has no other solution. 5. NUMERICAL
ILLUSTRATION
We present in this section some numerical simulations in order to illustrate the theoretical results given in the above sections. The approximate solutions shown below have been computed using classical finite-difference algorithms for steady-state problems (the nonlinear discrete equations, which include finite-difference approximations of the boundary conditions, are solved using a Newton-like method). The discretization step, however, has been kept small enough so that the error due to discretization is negligible with respect to the error due to the truncation of domain.
5.1. Comparison of different truncated solutions The first set of examples compares the convergence of solutions obtained with the different boundary conditions. More precisely, for each of the boundary conditions under consideration [that is, for problems (2.21), (2.22), (3.42) and (4.1)], we show in Figs l-5 the approximate solution computed on the interval (-1, l), together with the “exact” solution (i.e. a truncated solution computed on a much larger interval). Here the ignition temperature is 0 = 0.5 and g(s) has the expression: g(s) = K(s - OS)(l - s), (5.1) for 0.5 5.~51.
Error
0 -1.0
-0.8
-0.6
estimates
-0.4
Fig. 1. The solution
for travelling
-0.2
-0.0
0, of problem
1195
wave solutions
0.2
0.4
0.6
0.8
1.0
(2.21) (a = 1, C, = 1.8485).
0 -1.0
-0.8
-0.6
-0.4
Fig. 2. The solution
-0.2
-0.0
U, of problem
0.2
0.4
0.6
0.8
1 .o
(2.22) (a = 1, c1 = 1.8430).
0 -1.0
-0.8
-0.6
-0.4
Fig. 3. The solution
-0.2
-0.0
ti, of problem
0.2
0.4
0.6
0.8
(3.42) (a = 1, ?, = 1.8430)
1 .o
R. FERRETTI and B. LARROUTUROU
1196
0
-1.0
-0.8
-0.6
-0.4
Fig. 4. The solution
-0.2
-0.0
u, of problem
0.2
0.4
0.6
0.8
1 .o
(4.1) (a = 1, c1 = 1.8456).
0 -1.0
-0.8
-0.6
-0.4
Fig. 5. The “exact”
-0.2
solution
-0.0
0.2
of problem
0.4
0.6
0.8
1.0
(2.15) (c = 1.8446).
In order to compare these different solutions, we have also solved the bounded-domain problems on a larger interval (with a = 2), and the difference (in absolute value) between the two approximate solutions obtained in (-1, 1) and in (-2,2) has been plotted over the smallest interval, thus giving an idea of the different rates of convergence (see Figs 6-9). Notice that the scale on the y-axis may be different from one figure to another. The main conclusion here is that, even for such small values of the parameter a, the qualitative behaviour of these plots perfectly matches the theoretical results, and the solution of problem (4.1) with the mixed condition converges faster. 5.2. Nonuniqueness of the solution with the Neumann condition We give here an example of the lack of monotonicity for the function _J? defined in proposition 3.3 [as pointed out by remark 3.6, this means a lack of uniqueness for the solution of problem (3.42)]. Here, g(s) is the piecewise linear function shown in Fig. 10. Figures 11 and 12 show the behaviour of U, for two different values y1 and y2 such that y1 < y2 and g(y,) > 2(y,), which shows that 8 is not monotone increasing (actually, we see on Figs 11 and 12 that the solution uy, “does not see” the high values of g, whereas these high values of g are the reason why uJz takes much higher negative values).
Error
estimates
for travelling
1197
wave solutions
20.0 16.0
16.0 14.0 12.0 10.0 6.0 6.0 4.0 2.0 0.0 -1.0
-0.6
-0.6
-0.4
Fig. 6. The difference
-0.2
-0.0
0.2
Iii, - ti,l x IO-’ [solutions
0.4
0.6
of problem
0.6
1.0
(2.21)].
70.0 63.0 56.0 49.0 42.0 35.0 26.0 21.0 14.0 7.0 0.0 0
Fig. 7. The difference
\u, - u2\ x 1O-2 [solutions
of problem
(2.22)J.
20.0 16.0 16.0 14.0 12.0 10.0 6.0 6.0 4.0 2.0 0.0 -1.0
-0.6
-0.6
-0.4
Fig. 8. The difference
-0.2
-0.0
ILi, - ti,l x lo-’
0.2
[solutions
0.4
0.6
of problem
0.6
(3.42)].
1.0
R. FERRETTI and B. LARROUTUROU
1198
1 .o 0.9
I
I
I
I
0.8 _ 0.7
I
1
i
0.5 0.4 0.3 0.2 0.1
I
I
I I
I I
I
I
! 1
I
I
1
I
1
-0.0
-1.0
1
I
-0.8
i
I
-0.6
-0.4
Fig. 9. The difference
I
-0.2
I
-0.0
I ! I
I
I
I
1
0.2
0.4
lul - u2/ x 10v2 [solutions
1
0.6
of problem
I
0.8
1.0
(4.1)]
9(S) I
I
1
I
I
I
I
I
I
I -0.0
I
I
0.1
0.2
0.40
I
I
I
I
1
I
0.3
Fig. 10. The function
-0.00
I
I
0.80
1.20
Fig. 11. The solution
I I
I
I
I I
I
I
I
0.4
0.5
0.6
0.7
0.8
0.9
g for the solutions
1.60
2.00
U, of the forward
2.40
1.0
of Figs 11 and 12.
2.80
3.20
3.60
4.00
IVP (2.20) for y = y, = 0.1.
Error
-0.00
0.40
0.80
estimates
1.60
1.20
Fig. 12. The solution
for travelling
2.00
,
I
I
I
I/ I
0
-3.0
I
-2.4
I
I
I
I
l/r L
-1.8
3
I
-2.4
3.20
L
L
1
/
-0.0
0.6
’
I
I
I
1.2
-1.8
Fig. 13. Two solutions
L
-1.2
-0.6
ti, of problem
-0.0
I
1
I
-0.6
I
4.00
I’
/I
I -1.2
3.60
1.8
I
2.4
I
I -3.0
,
I
2.80
IVP (2.20) for y = yz = 0.5.
1
I
t
0
2.40
u, of the forward
1199
wave solutions
3.0
I
I 0.6
I
1.2
1.8
I
2.4
I
3.0
(3.42) with a = 3 and S, = 0.92 and 0.076.
R. FERRETTI and B. LARROUTUROU
1200
5.3. Divergence of truncated solutions The last example refers to the situation outlined in remark 3.8. In this case, we take 0 = 0.5 and g is given by: g(s) = Ks(s - 0.5)2(1 - S), (5.2) for 0.5 5 s i 1. Thus, g’(P) = 0. Figure 13 shows two approximate solutions of problem (3.42) on the interval (-3,3); these two solutions clearly correspond to the two behaviours described in remark 3.8: the first one is meaningful, whereas the second one is converging to the constant 8. 6. CONCLUSIONS
We have derived exponential convergence estimates for the solutions of several truncateddomain problems which are classically used for the approximation of the steady planar flame problem (1.1). These estimates do not hold irrespective of the boundary conditions which are imposed at both ends of the truncated domain; for instance, using a Dirichlet condition instead of a zero-flux boundary condition at the cold boundary x = -a gives less accurate results. In particular, we have shown that imposing at the burnt-gases truncated boundary (i.e. at x = a) a mixed boundary condition which mimics the asymptotic behaviour of the exact solution of (1.1) accelerates the convergence of the bounded domain solution to this exact solution. Moreover, it is our experience that using this new condition does not make the numerical solution of the bounded-domain problem more difficult. We therefore hope that this mixed boundary condition can be used to improve planar flame computations with more general models, including complex chemistry. REFERENCES 1. BERESTYCKI H. & LARROUTUROU B., A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. Reine Angew. Math. 396, 14-40 (1989). 2. BERESTYCKI H. & LARROUTUROU B., On planar traveling-front solutions of reaction-diffusion problems, INRIA Report (to appear). 3. BERESTYCKI H. & LARROUTUROU B., Mathematical modelling of planar flame propagation, in Reseurch Notes in Mathematics. Pitman-Longman, London (1990). 4. BERESTYCKI H., LARROUTUROU B. & LIONS P. L., Multi-dimensional travelling wave solutions of a flame propagation model, Archs ration. Mech. Analysis 111, 33-49 (1990). 5. BERESTYCKIH., NICOLAENKO B. & SCHEURER B., Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Analysis 16, 1207-1242 (1985). 6. BUCKMASTER J. D. & LUDFORD G. S. S., Theory of Laminar Flames. Cambridge University Press, U.K. (1982). 7. CLAVIN P., Dynamic behavior of premixed flame fronts in laminar and turbulent flows, Prog. Energ. Comb. Sci.
11, l-59 (1985). 8. GHILANI M. & LARROUTUROU B., Upwind
computation of steady planar flames with complex chemistry, Mod. (1991). 9. LARROUTUROU B., A conservative adaptive method for unsteady flame propagation, SIAM J. Sci. Stat. Camp. 10,
Math. Analysis Num. 25, 67-92 742-755 (1989).
10. LARROUTUROU B., Introduction 11. MARION M., Etude mathematique
to Combustion Modelfing. d’un modele
de flamme
Springer Series in Computational Physics (to appear). laminaire sans temperature d’ignition: I-Cas scalaire,
Ann. Fat. Sci. Toulouse 6, 215-255 (1984). 12. SERMANGE M.,
Mathematical
Opt. 14, 131-154(1986).
and numerical
aspects of one-dimensional
laminar
flame simulation,
Appl. Mufh.
Vol. 17, No. 12, pp. ,177-1200,
& Appl~ot,ons.
ERROR ESTIMATES FOR TRUNCATED OF TRAVELLING WAVE SOLUTIONS
0362-546X/91 $3.00+ .OO “G 1991 Pergamon Press plc
,991.
DOMAIN APPROXIMATIONS OF A COMBUSTION MODEL
R. FERRETTI Dipartimento di Matematica, Universita di Roma “La Sapienza”, 00185 Rome, Italy
and B. LARROUTUROU Cermics, INRIA, Sophia-Antipolis, (Received 20 September
06560 Valbonne, France
1990; received for publication
13 March 1991)
Key words and phrases: Travelling waves, error estimates, combustion models.
1. INTRODUCTION THE MOST fundamental
problem in combustion theory concerns a steady planar pre-mixed flame in an infinite tube. In the simplest case where one assumes a one-step chemical mechanism and a unit Lewis number and when one uses the classical isobaric approximation, the steady (or travelling-front) solution of the reacting flow equations is described by the following simple model (see e.g. [3, 6, 7, lo]): in IR,
--uM + CU’ = g(u) i U(---co) = 0,
u(+w)
= 1.
(1.1)
Here, the C?’ function u is the normalized temperature defined on all of IR; 0 (resp. 1) is the normalized temperature of the fresh mixture (resp. of the burnt gases) at the left (resp. right) end of the infinite tube. The unknowns in (1.1) are both the temperature field U(X) and the positive scalar c, which measures the normalized gaseous mass flux across the steady flame: in some sense, c plays the role of an eigenvalue in this problem. In the differential equation (1.1) the terms u”, CU’ and g(u) represent respectively heat conduction, heat convection by the flow, and heat source released by the chemical reaction. We will assume that the function g has an ignition temperature 8 [see (2.11) below]. For numerical purposes, one has to approximate the solution (u, c) of (1.1): one considers a bounded domain [-a, a] (with a “large enough”) and solves for a truncated approximation of (1 .l). Several such bounded-domain problems are considered by different authors (see, e.g. [8, 9, 12]), all being of the following form: one seeks for u, E (?‘(-a, a) and c, E IF?satisfying: -u; subject
to one of the following
to one of the following
+ c, U:, = g(u,)
boundary
conditions
-U:, (-a)
+ c, u, (-a)
boundary
conditions &(a)
= 0
in (-a,
a),
(1.2)
at x = -a:
= 0
or 2.4,(-a) = 0,
(1.3)
at x = a: or u,(a) 1177
= 1,
(1.4)
1178
and to the additional
R. FERRETTIand B. LARROUTUROU
condition: u,(O) = 8.
(1.5)
Condition (1.5) is added in order to keep c, unknown in (1.2)-(1.5) (see the discussions of this point in [l, 51). Some of the problems (1.2)-(1.5) [depending on the boundary conditions (1.3) and (1.4) which are actually used] have been studied from a mathematical point of view. One can then show that the solution (u,, c,) of (1.2)-(1.5) converges as a tends to +w to a solution (u, c) of (1.1) which in addition satisfies u(O) = 8. In fact, investigating a bounded-domain problem of type (1.2)-(1.5) and studying the limiting behaviour as a tends to +co is a good way of proving the existence of a solution to the original problem (1.1) (see [2-5, 111). Because the existing convergence proofs rely on compactness arguments, they do not provide any estimate of the errors Ic, - cl and IIu, - ~11. It is precisely our purpose in this work to derive such error estimates, whose interest for numerical calculations is clear. Under the appropriate conditions on the nonlinear function g, we will prove that the convergence of there exist a positive constant K and an explicitly known positive (u, 9c,) to (u, c) is exponential: constant k such that, for a large enough, the following holds: Ic, - cl I KePka,
IIu, - ~ll~z~-~,~)I Keek”.
(1.6)
The precise hypotheses and results are stated in theorems 3.1, 3.2, 3.6 and 4.2 below. The paper is organized as follows. We first recall, in Section 2, some known results about problem (1.1). The new mathematical results, and in particular error estimates of the form (1.6) are presented in Sections 3 and 4. Our main result is presented in Section 4, where we show that using a new boundary condition at x = a [instead of one of the conditions (1.4)] accelerates the convergence of the corresponding truncated solution to (u, c) (see theorem 4.2 below). Lastly, we illustrate the various convergence (or divergence) results by showing some numerical examples in Section 5. 2. SOME
We recall in this section
several known
KNOWN
results
RESULTS
about
2. I. A comparison principle We begin with a comparison principle for solutions with (1.1). This result was proved in [2, 31.
problem
(1.1) and related
of initial value problems
questions.
(IVP) associated
PROPOSITION 2.1. Let g, and g, be two Lipschitz-continuous functions, and let x0, 6, cl, c2, 01~,CY~ be given real numbers. For i = 1,2, consider the solutions ut and u2 of the forward IVP:
-Ui’
I
+ CjUi' = gi(Uj)
ui(xlJ = 6,
If:
g1
5
on [x0, +w),
&(x(J) = a;. g2
on
14
+a),
Cl zc21
011 2 cY2> 0, then ut (x) r u,(x)
for any x > x,, such that u; > 0 and ui > 0 on [x0, x).
(2.1)
(2.2)
(2.3) (2.4)
Error estimates
for travelling
1179
wave solutions
Remark 2.1. The proof of this result, which we do not present in detail here, relies on inverting the functions ui and u2 in the intervals where they are monotone increasing. Setting x;(s) = u;l(s) (i.e. i(;(xi(s)) = s) for i = 1, 2 and for s in some interval [a, S’), we can define Zi(S) = (d/ds)xi(s) and vi(s) = Zi(S)~l = u,!(xi(s)). Then, the functions yi satisfy:
(2.5) and one can show that y, 2 y, in [a, S’], whence [2, 31 for the details.
zi I z2, x, 5 x2 and u1 1 u2. We refer to
Remark 2.2. With the assumptions of proposition 2.1, one can also show that, if one of the two inequalities (2.3) or (2.4) is strict, then U,(X) > uZ(x) for any x > x0 such that u; > 0 and u; > 0 on [x0, x). We can also state a similar
result for the backward
IVP.
PROPOSITION 2.2. Let g, and g, be two Lipschitz-continuous functions, and let x0, 6, cl, c2, a,, 01~ be given real numbers. For i = 1,2, consider the solutions ui and u2 of the backward IVP: -Ul’ + ciu;
= g,(u,)
on (-~,xJ,
(2.6)
u; (x0) = a,.
i U;(%) = 6, If: g, 5
on
g2
cl 2
t-m,
(2.7)
61,
c2,
(2.8)
(2.9)
o
then U, (x) L u2 (x) for any x < x0 such that u; > 0 and U; > 0 on (x, x0]. 2.2. The basic results for problem (1.1) Let us now recall the main existence and uniqueness in the whole article that the function g satisfies:
result for problem
g is Lipschitz-continuous 3 0 E (0, l),
g = 0 on (-03,191,
in R,
g’(1) < 0. are classical
in combustion
(2.11) (2.12)
in (2.1 l), 19is an ignition temperature. Moreover, we will always assume, below, that g is differentiable at the point s = 1 and satisfies:
(2.10)-(2.13)
(2.10)
g > 0 on (0, l),
g = 0 on [l, fco);
All assumptions
(1.1). We will assume
except for theorem
2.3
(2.13) theory.
R. FERRETTI and B. LARROUTUROU
1180
Remark 2.3. All results presented g is allowed with:
to be discontinuous
below could be straightforwardly extended to the case where at 8, i.e. to the case where g is Lipschitz-continuous on (0, 11, (2.14)
lim g(s) > 0, Sh0 and satisfies
(2.1 l)-(2.12).
The following
result has been proved
THEOREM 2.3. There exists a unique
by Berestycki
solution
et al. [5].
(u, c) of:
-UI’ + CU’ = g(u)
in IR, (2.15)
i u(-co) and this solution
satisfies
= 0,
U(0) = e,
u(+oo)
= 1,
c > 0 and U’ > 0 in I?.
We omit the proof, which relies on a shooting (see [2, 3, 51). Let us simply add that the solution U(X) = t!Iecx
method and uses the comparison (u, c) of (2.15) satisfies:
for all x 5 0,
principle
(2.16)
whence: U’(0) = 09. We also recall the following
result about
the asymptotic
(2.17) behaviour
of U(X) as x tends to +a.
PROPOSITION 2.4. Assume that the hypothesis (2.13) holds. Then there exists a positive k such that: u’(x) = kremrx + o(emrx), U(X) = 1 - kemrx + o(ePrx),
constant (2.18)
as x tends to +co, with:
r=
Jc”
- 4g’(l)
- c
2
(2.19)
> 0.
The proof of proposition 2.4 relies on the investigation of the linearized --u” + CU’ = g’(l)(u - 1) and on some classical results of ordinary differential theory (see [2, 31). Throughout the paper we will always call (u, c) the unique solution of (2.15). 2.3. A related initial value problem We will consider several times in the sequel the following with (2.15). For any y > 0 we call U, the solution of:
L
2.l; +
yu; = g(u,)
u,(O) = 8,
u{(O) =
We already know from (2.17) that U, = u on IR,. follows from Then, the following result essentially to [2, 3, 51 for the proof).
initial
value problem
on R+,
associated
(2.20)
ye. the comparison
equation equations
principle
(we refer
1181
Error estimates for travelling wave solutions
PROPOSITION 2.5. If y > c, then u_l > 0 on R, and u,(+m) = +a~. In particular, there exists a unique positive real number x,, such that u.,(x,,) = 1. If 0 < y < c, then u,(+-co) = --co and there exists a unique positive real number .C, such that uC(Q = 0, with u,(&) E (f9, 1).
We will use below the classical fact that, for any x > 0, u,(x) and UC(X) continuously on y. We will also need the following property (see [2, 31). LEMMA
2.6. The solution
uy of (2.20) satisfies
u_l 5 y on the interval
lx E R, , u,(x)
depend
I
11.
2.4. Truncated solutions Finally, we end this section by considering some truncated solutions of (2.19, i.e. solutions of a problem which is similar to (2.15) but posed on a bounded domain (-a, a). Such solutions have been considered by several authors, either from the mathematical point of view or for the numerical approximation of problem (1.1) (see, e.g. [3, 5, 8, 121). In particular, two bounded domain approximations have mainly been considered in the literature. The first one consists of finding (ii,, co) satisfying: - 22; + i +:(-a) while the solution
in (-a, a),
coii,: = g(z7,) + C*,ii,(-a)
(u,, c,) of the second
problem
-24; +
c,u:, = g(u,)
u,(-a)
= 0,
L
a*(O) = 0,
= 0,
(2.21)
ii,(a) = 1,
satisfies: in (-a, a),
u,(O) = 8,
(2.22)
u,(a) = 1.
We emphasize that any confusion between the notation u, defined by (2.22) and the notation 11~defined by the IVP (2.20) should be avoided in the sequel. The following existence, uniqueness and convergence results are proved in [5] for problem (2.21) and in [2, 31 for problem (2.22). THEOREM 2.7. For any a > 0, there exists a unique solution (ii,, co,) of (2.21); satisfies ii: > 0 in (-a, a). When a tends to +m, ti, converges to u in C$,,,(lR) and C~ converges to c. THEOREM 2.8.
There exists Li:> 0 such that problem (2.22) has a unique only if a > CT;this solution satisfies U; > 0 in (-a, a). When a tends to fcu, u, converges to u in C&(R) and c, converges 3. EXPONENTIAL
ERROR
solution
this solution
(u,, co) if and
to c.
ESTIMATES
In this section, we will derive error estimates for three different truncated the solutions (ii,, c,) and (u,, c,) considered in Section 2.4 above.
solutions,
including
3.1, Zero-flux condition at the left boundary We begin with the simplest case, where a zero-flux condition is imposed at the left boundary x = -a, i.e. we consider the solution (ii,, CJ of (2.21). We will prove the following result.
R. FERRETTI and B. LARROUTUROU
1182
THEOREM 3.1. Assume that g satisfies (2.13) and is monotone decreasing in some left neighbourhood of 1. Then, there exist two positive constants a,, and K such that, for any a > a,, the solution (ii,, CJ of (2.21) satisfies the following estimates:
ICa- cl
I
Ke-““,
IIii, - uII~~~-~,~)5 Ke-‘“,
(3.1)
where r is given by (2.19).
Remark 3.1. The fact that g is decreasing
in some neighbourhood of 1 follows from assumption but does not follow from (2.10) to (2.13).
(2.13) if g is C?’ in this neighbourhood,
Proof. We divide the proof into four steps. Step 1. An upper bound
for C~ - c. It is easy to see that U,(X) = Be’“* for proposition 2.5, we see that (5, > c, that determined by the equation Q = a (we comparison principle then shows that ii, 2 (because x,, decreases as a function of y). Thus, we may invert ii, and define x,(s) z,(s) = x:(s), y,(s) = l/(z,(s)) = iiL(x,(s)). y(s) = l/(z(s)) = u’(x(s)). Then:
all x E [-a, 01, whence ii,: (0) = C~8. Then, from increasing, and that c0 is ii, = u,~ is monotone use here the notations of proposition 2.5). The u in [0, a], and that ?a decreases as a function of a by the relation Z&(X,(S)) = s, for s E [8, 11, and set We also define x(s) by u(x(.s)) = s, and z(s) = x’(s),
y:,(s) =
g(s) r, - Y, (s)’
(3.2)
Y,(B)
Ge,
(3.3)
= %(a).
(3.4)
also shows (see remarks
2.1 and 2.2) that ya (s) > y(s) for
y’(s) = c - ;+, Y(B) =
co,
Y(l) = 0, Since ?a > c, the comparison principle all s E [Q, 11; therefore, we have:
Y,(l)
=
(3.5) Integrating
this relation
over the interval y,(l)
[0, l] yields:
- ~(1) 2 (co - c)(l - 0) + Y,(O) - Y(O),
(3.6)
or equivalently: - -ciiiL(a). C,
(3.7)
Step 2. Exponential
estimate for i;, - c. Since ?a decreases as a function of a, there c0 I C?< +os for all a large enough. Now consider IVP:
-w; +
Ew:, = 0
exists an upper bound for ~~a: we have the solution w, of the following backward in (-co, a), (3.8)
i W,(Q) = 1,
WA(U) = &(a).
Error
The comparison Then, we get:
principle
estimates
for travelling
for backward
2.2) shows that w, 2 ii, on [0, a].
IVPs (proposition
a,:(4
u(x) 5 U,(X) 5 w,(x)
1183
wave solutions
= 1 + ~
?
(e
p(wpa)
- l),
for all x E [0, a]. On the other hand, we know from (2.18) that we can find a positive K such that: 0 5 1 - U(X) I Ke-‘” for all x 2 0. Writing
(3.9) for x = a - 1 and using (3.10),
constant (3.10)
we can write:
G(Q) -c^ - 1) 2 u(a - 1) L 1 - Ke-r(a-l), (e c
l+p
(3.9)
(3.11)
and an estimate of the form c0 - c 5 Ke-‘” follows from (3.7) (here and in the sequel, the notation K denotes a positive real number which does not depend on a, but which may not be the same at each time it appears). Step 3. Exponential estimate for ii, - U. Let us deduce from the previous step an estimate on ii, - U. We already have from step 1 an estimate on y, - y: since y, - y is increasing on the interval [f?, 11, we have: 0 I y,(s)
- y(s) I y,(l)
5 Ke-‘“,
- y(1) = a:(a)
(3.12)
for all s E [e, 11. Starting from this inequality, we are going to successively derive estimates on z - z,, x - x, and lastly on ii, - U. Let s0 E [0, l] be a fixed number, to be adequately chosen later. For s E [e, s,], we have:
0 5 44 - z,@) =
Y,(S) - Y(S) I y(s)y
(s)
a
Y,M
- Y(S)
min y(Q2’ B
(3.13)
so that there exists K > 0 such that: 0 I
Integrating
this inequality
z(s) - z,(s) 5 Ke-'"
for all s E [0, so].
(3.14)
for all s E [0, s,].
(3.15)
on (0, s), we get in turn:
0 I
x(s) - x,(s)
Beside this, we know from lemma
I
Ke-'"
2.6 that ii; I Ca I E. Then,
0 I &(X(S)) - U(X(S)) = ii, (x(s)) - i&(x,(s)) from which we deduce
for any s E [e, s,], we have:
I @x(s) - x,(s)),
(3.16)
that: 0 I ii,(x)
- U(X) % Ke-‘”
v x E 10, X@,>l~
(3.17)
Now, it remains to show that the same inequality holds for all x E [x(+,), a]. But we already know from (3.10) that ii,(a) - u(a) I Ke-‘“. Let us therefore examine whether the difference ii, - u can have a local maximum at a point xi in the open interval (x&J, a). If this is the case, we have: ii; I U”(X,), (3.18) ii,’ = u’(x,), 0, (x1) > U&I) > so 7
R. FERRETTI and B. LARROLJTUROU
1184
which implies g(ii,(x,)) 2 g(u(xi)). that g is decreasing on the interval
If, from the outstart, we have chosen s, large enough [s,, , 11, we get a contradiction. Therefore, we obtain:
0 I ii, - u 5 Ke-‘” The same property known.
obviously
Step 4. Exponential To end the proof, differential equations
holds on the interval
on [0, a]. [-a,
so
(3.19)
0] where both u and ii, are explicitly
estimate for ti: - u’. it remains to show that the estimate (3.1) holds in C2 norm. From satisfied by u and ii,, we see that we just have to check that:
the
la;(x) - u’(x)1 I Ke-‘” (3.20) for all x E [-a, a]. Firstly, checking (3.20) for x E [-a, 0] is obvious. Also, arguing as we did above to deduce (3.17) from (3.1.5), we see from (3.15) that (3.20) holds for allx E [0, x(s,)]. Beside this, we have: Iii:(a)
- u’(a)\ I Ke-‘”
(3.21)
from step 2 above and (2.18). Thus, it remains to examine the case where the difference ii; - U’ is extremal at some point xi in the open interval (x&J, a). In such a case, we have u”(x,) = ii:( and we get: ]COi$(Xi) - CU’(Xi)I = g@(Q)
- &%(x1))
from (3.19) and because g is Lipschitz-continuous. But it is easy to deduce (ii; - u’(x,)l I Ke-‘“, which concludes the proof. n 3.2. Dirichfet condition at the left boundary Let us now come to the solution (u, , c,) of problem
(3.22)
5 Ke-‘“,
from (3.22) that
(2.22). We will prove the following
result.
THEOREM 3.2. Assume that g satisfies (2.13) and is e’ in some left neighbourhood of 1. Then, there exist three positive constants a,, K and r’ such that, for any a > a,, the solution (u, , c,) of (2.22) satisfies the following estimates:
Ic, - cl 5 Ke-““,
(Iu, - z.JII~+~,~)s Ke-““.
(3.23)
Remark 3.2. Carefully
examining the proof of theorem 3.2 below, one can see that the constant r’ in (3.23) is smaller than r given by (2.19), and that for any E > 0, r’ can be chosen in the interval (i - E, ?) where F = min(r, c), provided that a, is taken large enough. Therefore, the “asymptotic rate of exponential convergence” F of the truncated solution (u, , c,) is explicitly known, and may be smaller than the rate of exponential convergence r of the truncated solution (ii,, Q) of (2.21).
Proof. We divide it into two steps. Step 1. Exponential estimate for c, - c. Let a be large enough so that problem (2.22) has a solution
(u, , c,). For x < 0, u,(x) is given by:
e=,x _ e-V U,(X) = e
’
(3.24)
> c,e.
(3.25)
1 _ e-c,a
and we have:
Error
estimates
for travelling
Let (ii,, CJ be the solution of (2.21). If we suppose comparison principle (proposition 2.1 and remark Therefore, we know that c, I co. On the other hand, if we assume that:
1185
wave solutions
that c, 2 c~,, we get u:(O) > ii:(O), and the 2.2) yields u,(a) > ii,(a), a contradiction.
(3.26) then u:(O) 5 u’(0) and c, < c. The comparison principle then another contradiction. Thus, (3.26) does not hold and we obtain:
implies
that
u,(a)
c(1 - e-caa) < c, 5 Co.
< u(a),
(3.27)
Since theorem 2.8 shows the existence of a lower bound on c, (c, > c0 > 0), and using theorem 3.1, we obtain: c - ce-‘@ I c, 5 c + Ke-‘“, (3.28) which proves the desired
estimate
for Ic, - cl.
Step 2. Exponential estimate for u, - U. If c, L c, then the exponential estimate for u, - u can be derived using exactly arguments as in step 3 of the proof of theorem 3.1. Thus, it remains to prove a similar in the case where c, < c. Assume therefore that c, < c. Define the functions x, and y, by u,(x,(s)) u; (x0 (s)) = ya (s) for all s E [0, 11. We will also use the functions x and y defined in of theorem 3.1. We already know from (3.27) that: 4 (0) = Y,(B) > Y(B) = u’(O),
the same estimate = s and the proof
(3.29)
and that: Y,(l) We claim that these inequalities
(3.30)
> Y(1) = 0.
imply that: on [0, 11.
Y, > Y
(3.31)
Indeed, if y,(s) = y(s) for some s E (0, l), then the difference y;(s) - y’(s) is negative (from the differential equations satisfied by ya and y), and the inequalities (3.29)-(3.30) cannot hold together (notice that the comparison principle cannot be directly used since c, < c). Now let s, E [0, l] be a fixed number, to be adequately chosen later. Since we know that: Y:, (4
- Y ‘(4 =
G -
c+
m
Y,(S) -
Y(4 Yn @lY@) ’
(3.32)
we obtain: Y;(s) - Y’(S) 5 R(Y,@) - Y(S)) for all s E [e, s,], with R=
max g(s) e
~(4~ ’ 0
(3.33)
R. FERRETTIand B. LARROUTUROU
1186
Using Gronwall’s
lemma,
we get: r,(s)
.d@leR’“-o’
- Y(S) 5 [u, (0) -
(3.34)
for all s E [0, s,]. But it is easily seen from (3.25) (3.28) and (3.29) that y,(B) - y(B) 5 Ke-“” for some constant r’ independent of a, and we can write that: 0 < y, - y 5 Ke-“”
(3.35)
on [e, ~1.
Now, arguing as in the proof of the comparison principle (see remark 2. l), one can deduce from (3.31) that u, > u in (0, a). Then, using the same arguments as in step 3 of the proof of theorem 3.1, we can prove that: u,(x)
- u(x) I Ke-“”
for all x E [0, x&J].
(3.36)
Since 0 I u,(a) - u(a) 5 KC”“, it remains to examine the case where the difference u, - u has a local maximum at some point xi in the open interval (x(sO), a). If this happens, we have: u, (x1) > u(x,) > so 3
ul(xJ
U:,(Xl) = u’(x,),
5 U”(X,),
(3.37)
which yields: c,u’(x,) If, from the outstart,
- g(u,(x,))
5 cu’(x,)
we have chosen so close enough
(3.38)
- g(u(x1)). to 1 so that:
max g’(s) < 0 s,5s51 (which in particular
implies
that g is decreasing
0 I g(u(x,))
- g(u,(x,))
(3.39)
on the interval 5 (c - c,)u’(x,)
[so, 1]), we get: 5 Ke-“”
(3.40)
(we recall that u’ is uniformly bounded from lemma 2.6). Since the inverse of g is Lipschitzcontinuous in the interval under consideration from (3.39), we deduce from (3.40) that u,(xJ - u(xJ 5 Ke-““, whence: 0 I u,(x) The end of the proof
- u(x) 5 Ke-“”
is then analogous
to step 4 of the proof
3.3. Neumann condition at the right boundary We now wish to examine the following problem, atx = a: --a; + &,ti,: = g(ti,) in (-a, ( A,:(-a)
+ &z&(-a)
for all x E [0, a].
= 0,
of theorem
with a homogeneous
(3.41) 3.1.
n
Neumann
condition
a), C,(O) =
8,
z&(a) = 0.
(3.42)
We will see below that additional difficulties related to the behaviour of g in the neighbourhood of B occur for error estimates of the form (1.6) to hold for (ti,, &). We first need to investigate the existence and uniqueness of (ti,, 2J since no result like theorem 2.7 is known about this problem in the literature. It is easy to see that any solution (a,, &) of (3.42) satisfies: Z&(X) = e&X (3.43) for all x E [-a, 01, whence C;(O) = &0. Using the notations of proposition 2.5, we see that ti, = Uf,, and that solving problem (3.42) amounts to find & < c such that $, = a. This is why we need to prove some properties of the application 8: y - .?? defined in the interval (0, c). This is the object of the next two results.
Error estimates for travelling wave solutions PROPOSITION 3.3. The application
in the interval
8: y H k, (with L??defined
1187
in proposition
2.5) is continuous
(0, c) and satisfies: lim& Y/‘C
= +w.
(3.44)
Proof. Notice first that proposition 2.5 implies that, for all y E (0, c) and for all x > 0 with u;(x) has the sign of the difference ,?,, - x. xzg7,, Now, we argue by contradiction. Suppose that x is not continuous at some point y0 E (0, c). Assume that there exists a sequence (y,) with lim yn = y0 such that lim j$,, > &,. Then, n-+m n-+m for n large enough and for some x0 > &,, we have iYy, > x0. But we have $(x0) > 0 for all n large enough, whence u&(x,) 2 0 by the continuous dependence of uy with respect to y, and this contradicts the fact that x0 > &,. Arguing in the same way, we also get a contradiction if we assume that there exists a sequence (y,) with nkmm yn = y,, such that lim gYy,< iYo. Therefore, the application 2 is continuous. n++C= The argument for (3.44) is really the same. If there exists a sequence (y,) with y, < c and lim yn = c such that lim k,,” < +co, then we choose x0 > 0 so that gTy, < x,, for all n and we n-+m a-+m get ub,(xJ < 0 for all n, whence uL(xO) I 0, a contradiction. n PROPOSITION 3.4. Assume
then the application
that g has a right first derivative X: y ++ jZ7 satisfies:
g’(P)
at the point 8. If g’(P)
lirn,?? = ’ 2@@7’ YLO If g’(0’)
> 0,
(3.45)
= 0, then: lim$ YhO
= + co.
(3.46)
Proof. Notice
first that it follows from the comparison principle (proposition 2.1) that = max u,(x) is a monotone increasing function of y E (0, c) (see [2, 31 for the details). X>O Choosing a fixed y. E (0, c), we set so = ZL,,(&J E (0, 1). Consider first the case where g’(P) > 0. We can then find two positive constants m and M such that: m(s - e) < g(s) < M(S - e) (3.47) u~(,Q
for all s E (0, so]. Assume
that y < min(2&,
yo). Then,
consider
the solution
wy of:
e),
w:,(&)
which is given by: WY= e + (24,(2?) - e)exp(Y(x with a,”
= JM
2 9Y))(c0s]c0r(x
- i?)] - &sin[wr(x
- L,)]),
(3.49)
- y2/4. Setting: _Y,=Z?7+$[arctan(3
we see that w,(y,)
(3.48) = L$(.q) = 0,
- 7r],
= 8 and that wy > 8, w; > 0 in the interval
(y,, 2.J.
(3.50)
R. FERRETTIand B. LARROUTUROU
1188
Now, we claim that w, s uy on the interval [y,, &I. In fact, this does not directly follow from the backward comparison principle (proposition 2.2) since w~(Q = uk(.Q = 0. But we can notice that w, 5 uy in some left neighbourhood of & since w,“(&,) < us from (3.47) and our choice of y. Therefore, if there exists x = maxly E [y,, Q, w,(y) > u,(y)), we necessarily have w?(x) = u,,(x) and w;(x) 5 u;(x); we even have the strict inequality w;(x) < u;(x) since the identity w;(x) = u;(x) would imply w;(x) < u,“(x) and wr < uy in some left neighbourhood of x. Then, we can apply the forward comparison principle (proposition 2.1 and remark 2.2) to get a contradiction. Therefore, we have 8 = w,(y,) 5 u,(y,), which shows that yy 2 0, whence: (3.51) In a completely
similar way, for y < min(2&, -u,” +
yus,=
yO), we can consider the solution uy of:
m(u, - e),
(3.52) and define: .z,=a+$[arctan(F) with CIJ,”= w. that
-7r],
Then we can show that u, 5 u, on the interval
(3.53)
[z,, ZJ,
which implies (3.54)
Now, (3.54) shows that 2;;,is uniformly bounded as y tends to 0: we claim that this implies that: lim u,(&,) = 19. (3.55) Y’O Indeed, for any sequence (y,) with ,,li_mm yn = 0, we can extract a subsequence (still denoted (y,J) such that the sequence (&) converges to some positive limit x0. Then, we obtain lim uJ&,) = uo(xo) = 19since the functions u,, are equicontinuous from lemma 2.6, and n++CC (3.55) follows. We can now conclude the proof. Using (3.55) we see that, for any y small enough, we can choose two constants m7 and M, such that (3.47) holds for all s E (0, K,(Q], and so that: lim m, = ;y My = g’(0’). L YLO
(3.56)
Then, examining the limits as y tends to 0 of the inequalities: -&[ 71- arctan( % 7
ai,r&[n-arctan(22)],
which are the analogues of (3.51) and (3.54), we easily obtain (3.45). Lastly, in the case where g’(B+) = 0, we use the same construction
(3.57)
with lim M, = 0, and
(3.46) follows from examining the limit of the left inequality (3.57) as y ten&‘:0 0.
n
Error
estimates
for travelling
1189
wave solutions
Remark 3.3. As shown by the previous proof, the conclusions of proposition 3.4 could be reached from a more heuristical point of view by considering the linearization of the IVP (2.20) in the neighbourhood of the point (x = 0, uy = 0). Remark 3.4. discontinuous
we can
In connection with remark 2.3, at 0 and satisfies (2.14), we have lim& YLO
add
that,
in the
case
where
(3.58)
= 0.
To see this, it suffices to use the same construction lim m, = +oo; (3.58) then follows from examining YbO as y tends to 0.
g is
as in the proof of proposition 3.4, with the limit of the right inequality (3.57)
Remark 3.5. We already said that the application y E (0, c) - u,(&) is monotone increasing in monotone the interval (0, c). However, the application x: y - & itself is not necessarily increasing in (0, c) (see Section 5 below for an example where x is not monotone). Propositions
3.3 and 3.4 immediately
provide
an existence
COROLLARY3.5. Assume that g has a right first derivative Then, there exists a positive constant ci such that problem if and only if a L ri. Proof. As already
said, solving
problem
(3.42) amounts
&, = g(&) It is clear from the properties of the application only if a 2 ci = min 2(y) > 0. n YE[O,c)
result for pr’oblem (3.42).
g’(0’) at the point 8, with g’(B+) > 0. (3.42) has (at least) a solution (6,) to)
to find & < c such that
= a.
(3.59)
2 that the equation
(3.59) has a solution
if and
Remark 3.6. It is clear from the preceding proof and from the properties of _$?that proving a uniqueness result for problem (3.42) would be equivalent to proving the monotonicity of the application 2. Thus, the solution of (3.42) is not unique in general (see remark 3.5 above). We are now able to derive the following
error estimate
for problem
(3.42).
THEOREM 3.6. Assume that g satisfies (2.13) and is monotone decreasing in some left neighbourhood of 1. Assume moreover that g has a right first derivative g’(@) at the point 0, with g’(0’) > 0. Then, there exist three positive constants ao, K and r’ such that, for any a > a,, any solution (a,, 2,) of (3.42) satisfies the following estimates: Ii& - cl 5 Ke-““,
I/z?, - ul(,z(_,,,)
I Ke-““.
(3.60)
Remark 3.7. Carefully examining the proof of theorem 3.6 below, one can see that the constant r’ in (3.60) is smaller than r given by (2.19), and that, for any E > 0, r’ can be chosen in the interval (r - E, r) provided that a, is taken large enough. Therefore, the “asymptotic rate of exponential convergence” of the truncated solution (a, , 2J is the same as the rate of exponential convergence r of the truncated solution (ii,, Q of (2.21).
1190
R. FERRETTIand B. LARROUTUROU
Proof. In the sequel, (fi,, i$) denotes any solution of (3.42) for a large enough. It is already clear from (3.59) and proposition 3.3 that & converges to c (and therefore that li, converges to u in e,‘,,(R)) as a tends to +a~. We divide the proof into two main steps which are similar to steps 1 and 2 of the proof of theorem 3.1. Step 1. An upper bound
for c - F, . For s E [0, fi, (a)], we define x(s), z(s), and y(s) as in the proof of theorem 3.1, and we now set &(X,(S)) = s, z,(s) = x;(s) and y,(s) = l/z,(s) = &(x,(s)). Then it is easy to see that y, < y, whence y’ - y: 2 c - ea on the interval [B, &(a)]. Integrating this inequality and using the values of y(0) and y,(8), we obtain: (3.61)
Y(fi, (a)) - Y, (fi, (a)) 2 (c - G)&i (a).
Now, as a simple consequence of De L’HBpital’s theorem, we deduce from the equation (3.2) satisfied by y that y is differentiable at the point 1, with y’(1) = -r, where r is given in (2.19). Hence there exists a positive constant K such that: y(s) 5 K(l Then,
- S)
(3.62)
for all s E [e, 11.
using the facts that z&(a) > 0 and y,(ti,(a)) c - F, 5 $(l
= 0, we obtain
from (3.61): (3.63)
- G,(a)).
Step 2. Exponential
estimate for c - &. Since & converges to c as a tends to +co, we can find a positive lower bound for &: there exists c” > 0 such that F, > ? for all a large enough. Then, choosing a fixed real number x0 > 0 and setting +(x0) = sO, we have: (3.64) ii, 2 so > 8, for all a large enough.
Beside this, there exists a positive constant
m such that, for all s E [so, 11: (3.65)
g(s) 2 m(1 - s). Consider
the solution
w, of the following
backward
IVP:
- wI( + cw; = m(1 - wU)
in (-03, a), (3.66)
w, (a) = & (a) >
w;(a) = 0,
which is given by: w,(x)
with:
r, =
=
1
_
(‘i ~~~)(r2er1+” - rler2cx-a)),
c--<.
r 2
2
=c+dzzi>o 2
(3.67)
(3.68)
Then it is easy to check that w: > 0 on the interval (0, a), and we can argue as in the proof of proposition 3.4 to show that li, 5 w, on [0, a]. From (3.64), this implies that w,(x,) > sO, which writes: (1 - s&2 - rl) (3.69) 1 - ii,(a) 5 r2erl(~O-a) _ rl e~2(w-~) ; using now (3.63), we obtain
the desired
estimate
for c - &.
Error
estimates
for travelling
1191
wave solutions
The end of the proof is now completely analogous to steps 3 and 4 of the proof of theorem 3.1, with the only difference that we now have CO< c, fi, < u, whereas we had C~ > c, ii, > u n in the context of theorem 3.1. We leave the details to the reader. Remark 3.8. If g’(Q’) = 0, it is clear from proposition 3.4 and corollary 3.5 that problem (3.42) has at least two solutions for all a large enough. One of these solutions converges to (u, c) and satisfies the error estimates (3.60), whereas one of these solutions has a completely different behaviour: as a tends to +co, C0 converges to 0 and fi, converges in C&(R) to the constant 8. 4. IMPROVED
We now consider solution of:
a fourth -u;
ERROR
bounded-domain
ESTIMATE
problem.
We will henceforth
call (u,, c,,) a
in (-a, a),
+ C,U,: = g(u,)
f -&(-a)
+ c,u,(-a)
= 0,
h(O) = 0,
2
i
u;(a) =
J
‘a - 4g2’(1) - ‘“(1
(4.1)
- u,(a)).
The mixed Dirichlet-Neumann condition at the right boundary x = a has been chosen in order to mimic the behaviour of the true solution u since we know from (2.18) that: Jc’ u’(x) lim x-+m 1 - u(x) = r =
- 4g’(l)
- c . ’
2
(4.2)
thus, the above mixed condition seems to be the best condition to impose at x = a in order to achieve the “transparency” of the boundary. We will see in fact that (u,, c,) converges faster to (u, c) than the other truncated solutions. 4.1. Existence Let us begin by proving the existence of a solution to (4.1). In the sequel, when problem (4.1) has a solution (u, , c,), we will set: r, = About
problem
Jc,z - 4g’(l)
- c, .
2
(4.1), we first prove the following
PROPOSITION 4.1. Assume that g has a right first g’(@) > 0. Then problem (4.1) has (at least) a solution B is defined in corollary 3.5. Moreover, for all n a> a&jzj’
(4.3)
result. derivative g’(@+) at the point 8, with (u,, c,) for all a L 6, where the constant
let (u,, c,) be any solution of (4. l), and call (ii,, ca) the solution of (2.21). Then there exists a solution (ti, , e,) of (3.42) such that the following inequality holds: F” < c, < Cg.
(4.4)
1192
R. FERRETTI and B. LARROUTUROU
Proof. Let a I 6, and call (fi,, &) a solution of (3.42), and (ii,, co) the solution of (2.21). On the interval (&, cJ, consider the application [the notation U, is still defined by (2.20)] CB:y-
u;(a) ( 1 - u,(a)
It is easy to see that the application
Jr” -
CJlis continuous
lim CR(y) < 0, yLPL1 Thus, there exists a real number c, (4.1) [here, the notation uC, is also Now, let (u,, c,) be any solution the mixed boundary condition and fore, we have ??(c,) > a. If
- 4g’(l) 2
- Y
(4.5)
>.
on (i$, co,) and satisfies:
lim CR(y) = +w. y/E,
(4.6)
E (2*, CT=,) such that aZ(c,) = 0, and (uC,, c,) is a solution of defined by (2.20)]. of (4.1); it is easy to see that u, = uC,, and to deduce from proposition 2.5 that c, < I?~, u,(a) < 1, u:(a) > 0. ThereTt “2@@5’
this shows from the properties of the application & E (0, c,) such that 2(&) = a, which concludes
8 (propositions the proof. n
Remark below).
for problem
4.1. We have no proof
remark
4.3
THEOREM 4.2. Assume that g satisfies (2.13) and is C?’ in some left neighbourhood of 1, and g”(1) exists. Assume moreover that g has a right first derivative g’(P) at the point 8, g’(P) > 0. Then, for any E > 0, there exists r’ E (r - E, r), where r is given by (2.19), and positive constants a, and K such that, for any a > (I~, any solution (u, , c,) of (4.1) satisfies following estimates:
that with two the
4.2. Error estimate The error estimate
of uniqueness
3.3 and 3.4) that there exists
for the solutions
(4.1) (see however
of (4.1) is given in the following
]c, - cl I Kem2”‘,
I/u, - u~~~+~,~~I Ke-2”a.
result.
(4.7)
Proof. Let E > 0. It follows from (4.4) and remark 3.7 that we can find r’ E (r - E, r) such that (notice that li, < U, < ii, in (0, a) from (4.4) and the comparison principle): ]c, - cl 5 Ke-““,
IIu, - uII~o~-~,~)5 Ke-““.
(4.8)
Therefore, our purpose is to prove that these estimates can be improved to yield (4.7). We are going to prove (4.7) in the case where c, > c. We leave it to the reader to check that the proof is identical in the case where c, < c (with the obvious modifications which come from the fact that U, - u is then negative). We will have two steps. Step 1. For s that y, y’(1) =
Exponential estimate for c, - c. E 10, U, (a)], we define as before x(s), z(s), and y(s) and x,(s), .zU(s), and ya (s). We know - y is positive and decreasing on the interval [6, u,(a)]. Moreover, we know that -r, and it follows from the existence of g”(1) and from De L’Hopital’s theorem that y
Error
estimates
is e2 in some left neighbourhood a positive constant K such that:
for travelling
of 1. Then,
1193
wave solutions
using the mixed boundary
condition,
we can find
0 5 lYa(4 64) - Y(%WI r(l - %7wl + (r - r,)(l - x7@))
5 IYe47(4) 5
K(1 - 2.4,(a))2 + K(c, - c)(l - u,(a)),
(4.9)
where r, < r is given by (4.3) (we have used the fact that the application y y -JY2 - 4&(l) 2 is decreasing
and Lipschitz-continuous).
Hence,
y,(u,(a))
Now, we have the analogue
- Y
using (4.8) and (2.18), we get: I Ke-2”a.
- y(u,(a))
(4.10)
of (3.5), that is: (4.11)
y;(S) - y’(s) 1 c, - c for all s E [e, u,(a)].
By integration
over this interval,
we obtain: (4.12)
(c, - c)u, (a) 5 Y, (u, (a)) - Y(%(@), which together
with (4.10) gives the desired
estimate
for c, - c.
Step 2. Exponential estimate for u, - u. Let se be chosen as in step 3 of theorem 3.1. Using the same arguments that: X.;;~~Ojl [u,(x)
- 441
as there, we can prove
5 Kee2”“,
(4.13)
and that u, - u cannot have a maximum in the open interval (x&J, a). Then examine the case where u, - u has a maximum at the point x = LI. If this happens, u;(a) 2 u’(a), which means that: Y, (u,(a))
it remains to then we have (4.14)
2 Y(U(Q)).
Since y is C?’ in the neighbourhood of 1 with y’(1) < 0, we can now choose E such that max y’(s) < 0. For a large enough, we have 1 - E I u(a) I u,(a) < 1, whence: s E [l--E, 11 Y(0)) Thus,
(4.14) and (4.15) show that y(u(a)) y(u(a))
Since the inverse
E [y(u,(a)),y,(u,(a))], - y(u,(a))
of y is Lipschitz-continuous u,(a)
(4.15)
2 Y(%(Q)). and (4.10) now implies:
I Ke-2r’a.
in the interval
(4.16) [l - E, 11, we obtain:
- u(a) 5 Kem2”‘,
(4.17)
which proves that: Kl - u 5 Ke-2r’a The end of the proof
is then similar
on [0, a].
to step 4 of the proof
of theorem
We end this section with the following additional result on the solution the sign of c, - c for the solution (u,, c,) of (4.1).
(4.18) 3.1.
H
of (4. l), which gives
R. FERRETTI and B. LARROUTUROU
1194 LEMMA 4.3.
large enough,
Assume that the hypotheses of theorem 4.2 hold and that g”(1) # 0. Then, the difference c, - c has the same sign as -g”(l).
Proof. A straightforward
calculation
and the use of De L’HBpital’s
y”( 1) = - $2
(4.19) for a large enough,
Y(% (a)) < r(l - u,(a)).
y ~
= r,(l
show that:
-#‘.
Assuming that g”(1) < 0, we therefore get y”(l) < 0. Then, to 1 and we can write (we recall that y’(l) = -r):
If c, I c, we havey(u,(a)) r _~,(u,(a)) this is impossible since the application
theorem
- u,(a)),
JYZ- W(l) -
for a
and we obtain
u,(a)
is close
(4.20)
r, < r from (4.20). But
Y
-I
is decreasing.
Remark 4.2. In the combustion increasing,
we have g”(1) = -f’(l)
context, where g is of the form g(s) = (1 - s)f(s) with f < 0. Thus, any solution (u,, c,) of (4.1) satisfies c, > c.
Remark 4.3. Let us also add that, if g’ is constant
in some left neighbourhood of 1, then for all a large enough (u, c) is the only solution of (4.1). Indeed, if g(s) = r(1 - s) in some interval [I - E, 11, there exists a constant k > 0 such that U(X) = 1 - keerx for all x large enough, so that (u, c) is a solution of (4.1) for a large enough. Moreover, we also have y(s) = r(1 - s) for all s E [l - E, 11, and we can argue as in the proof of lemma 4.3 to show that (4.1) has no other solution. 5. NUMERICAL
ILLUSTRATION
We present in this section some numerical simulations in order to illustrate the theoretical results given in the above sections. The approximate solutions shown below have been computed using classical finite-difference algorithms for steady-state problems (the nonlinear discrete equations, which include finite-difference approximations of the boundary conditions, are solved using a Newton-like method). The discretization step, however, has been kept small enough so that the error due to discretization is negligible with respect to the error due to the truncation of domain.
5.1. Comparison of different truncated solutions The first set of examples compares the convergence of solutions obtained with the different boundary conditions. More precisely, for each of the boundary conditions under consideration [that is, for problems (2.21), (2.22), (3.42) and (4.1)], we show in Figs l-5 the approximate solution computed on the interval (-1, l), together with the “exact” solution (i.e. a truncated solution computed on a much larger interval). Here the ignition temperature is 0 = 0.5 and g(s) has the expression: g(s) = K(s - OS)(l - s), (5.1) for 0.5 5.~51.
Error
0 -1.0
-0.8
-0.6
estimates
-0.4
Fig. 1. The solution
for travelling
-0.2
-0.0
0, of problem
1195
wave solutions
0.2
0.4
0.6
0.8
1.0
(2.21) (a = 1, C, = 1.8485).
0 -1.0
-0.8
-0.6
-0.4
Fig. 2. The solution
-0.2
-0.0
U, of problem
0.2
0.4
0.6
0.8
1 .o
(2.22) (a = 1, c1 = 1.8430).
0 -1.0
-0.8
-0.6
-0.4
Fig. 3. The solution
-0.2
-0.0
ti, of problem
0.2
0.4
0.6
0.8
(3.42) (a = 1, ?, = 1.8430)
1 .o
R. FERRETTI and B. LARROUTUROU
1196
0
-1.0
-0.8
-0.6
-0.4
Fig. 4. The solution
-0.2
-0.0
u, of problem
0.2
0.4
0.6
0.8
1 .o
(4.1) (a = 1, c1 = 1.8456).
0 -1.0
-0.8
-0.6
-0.4
Fig. 5. The “exact”
-0.2
solution
-0.0
0.2
of problem
0.4
0.6
0.8
1.0
(2.15) (c = 1.8446).
In order to compare these different solutions, we have also solved the bounded-domain problems on a larger interval (with a = 2), and the difference (in absolute value) between the two approximate solutions obtained in (-1, 1) and in (-2,2) has been plotted over the smallest interval, thus giving an idea of the different rates of convergence (see Figs 6-9). Notice that the scale on the y-axis may be different from one figure to another. The main conclusion here is that, even for such small values of the parameter a, the qualitative behaviour of these plots perfectly matches the theoretical results, and the solution of problem (4.1) with the mixed condition converges faster. 5.2. Nonuniqueness of the solution with the Neumann condition We give here an example of the lack of monotonicity for the function _J? defined in proposition 3.3 [as pointed out by remark 3.6, this means a lack of uniqueness for the solution of problem (3.42)]. Here, g(s) is the piecewise linear function shown in Fig. 10. Figures 11 and 12 show the behaviour of U, for two different values y1 and y2 such that y1 < y2 and g(y,) > 2(y,), which shows that 8 is not monotone increasing (actually, we see on Figs 11 and 12 that the solution uy, “does not see” the high values of g, whereas these high values of g are the reason why uJz takes much higher negative values).
Error
estimates
for travelling
1197
wave solutions
20.0 16.0
16.0 14.0 12.0 10.0 6.0 6.0 4.0 2.0 0.0 -1.0
-0.6
-0.6
-0.4
Fig. 6. The difference
-0.2
-0.0
0.2
Iii, - ti,l x IO-’ [solutions
0.4
0.6
of problem
0.6
1.0
(2.21)].
70.0 63.0 56.0 49.0 42.0 35.0 26.0 21.0 14.0 7.0 0.0 0
Fig. 7. The difference
\u, - u2\ x 1O-2 [solutions
of problem
(2.22)J.
20.0 16.0 16.0 14.0 12.0 10.0 6.0 6.0 4.0 2.0 0.0 -1.0
-0.6
-0.6
-0.4
Fig. 8. The difference
-0.2
-0.0
ILi, - ti,l x lo-’
0.2
[solutions
0.4
0.6
of problem
0.6
(3.42)].
1.0
R. FERRETTI and B. LARROUTUROU
1198
1 .o 0.9
I
I
I
I
0.8 _ 0.7
I
1
i
0.5 0.4 0.3 0.2 0.1
I
I
I I
I I
I
I
! 1
I
I
1
I
1
-0.0
-1.0
1
I
-0.8
i
I
-0.6
-0.4
Fig. 9. The difference
I
-0.2
I
-0.0
I ! I
I
I
I
1
0.2
0.4
lul - u2/ x 10v2 [solutions
1
0.6
of problem
I
0.8
1.0
(4.1)]
9(S) I
I
1
I
I
I
I
I
I
I -0.0
I
I
0.1
0.2
0.40
I
I
I
I
1
I
0.3
Fig. 10. The function
-0.00
I
I
0.80
1.20
Fig. 11. The solution
I I
I
I
I I
I
I
I
0.4
0.5
0.6
0.7
0.8
0.9
g for the solutions
1.60
2.00
U, of the forward
2.40
1.0
of Figs 11 and 12.
2.80
3.20
3.60
4.00
IVP (2.20) for y = y, = 0.1.
Error
-0.00
0.40
0.80
estimates
1.60
1.20
Fig. 12. The solution
for travelling
2.00
,
I
I
I
I/ I
0
-3.0
I
-2.4
I
I
I
I
l/r L
-1.8
3
I
-2.4
3.20
L
L
1
/
-0.0
0.6
’
I
I
I
1.2
-1.8
Fig. 13. Two solutions
L
-1.2
-0.6
ti, of problem
-0.0
I
1
I
-0.6
I
4.00
I’
/I
I -1.2
3.60
1.8
I
2.4
I
I -3.0
,
I
2.80
IVP (2.20) for y = yz = 0.5.
1
I
t
0
2.40
u, of the forward
1199
wave solutions
3.0
I
I 0.6
I
1.2
1.8
I
2.4
I
3.0
(3.42) with a = 3 and S, = 0.92 and 0.076.
R. FERRETTI and B. LARROUTUROU
1200
5.3. Divergence of truncated solutions The last example refers to the situation outlined in remark 3.8. In this case, we take 0 = 0.5 and g is given by: g(s) = Ks(s - 0.5)2(1 - S), (5.2) for 0.5 5 s i 1. Thus, g’(P) = 0. Figure 13 shows two approximate solutions of problem (3.42) on the interval (-3,3); these two solutions clearly correspond to the two behaviours described in remark 3.8: the first one is meaningful, whereas the second one is converging to the constant 8. 6. CONCLUSIONS
We have derived exponential convergence estimates for the solutions of several truncateddomain problems which are classically used for the approximation of the steady planar flame problem (1.1). These estimates do not hold irrespective of the boundary conditions which are imposed at both ends of the truncated domain; for instance, using a Dirichlet condition instead of a zero-flux boundary condition at the cold boundary x = -a gives less accurate results. In particular, we have shown that imposing at the burnt-gases truncated boundary (i.e. at x = a) a mixed boundary condition which mimics the asymptotic behaviour of the exact solution of (1.1) accelerates the convergence of the bounded domain solution to this exact solution. Moreover, it is our experience that using this new condition does not make the numerical solution of the bounded-domain problem more difficult. We therefore hope that this mixed boundary condition can be used to improve planar flame computations with more general models, including complex chemistry. REFERENCES 1. BERESTYCKI H. & LARROUTUROU B., A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. Reine Angew. Math. 396, 14-40 (1989). 2. BERESTYCKI H. & LARROUTUROU B., On planar traveling-front solutions of reaction-diffusion problems, INRIA Report (to appear). 3. BERESTYCKI H. & LARROUTUROU B., Mathematical modelling of planar flame propagation, in Reseurch Notes in Mathematics. Pitman-Longman, London (1990). 4. BERESTYCKI H., LARROUTUROU B. & LIONS P. L., Multi-dimensional travelling wave solutions of a flame propagation model, Archs ration. Mech. Analysis 111, 33-49 (1990). 5. BERESTYCKIH., NICOLAENKO B. & SCHEURER B., Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Analysis 16, 1207-1242 (1985). 6. BUCKMASTER J. D. & LUDFORD G. S. S., Theory of Laminar Flames. Cambridge University Press, U.K. (1982). 7. CLAVIN P., Dynamic behavior of premixed flame fronts in laminar and turbulent flows, Prog. Energ. Comb. Sci.
11, l-59 (1985). 8. GHILANI M. & LARROUTUROU B., Upwind
computation of steady planar flames with complex chemistry, Mod. (1991). 9. LARROUTUROU B., A conservative adaptive method for unsteady flame propagation, SIAM J. Sci. Stat. Camp. 10,
Math. Analysis Num. 25, 67-92 742-755 (1989).
10. LARROUTUROU B., Introduction 11. MARION M., Etude mathematique
to Combustion Modelfing. d’un modele
de flamme
Springer Series in Computational Physics (to appear). laminaire sans temperature d’ignition: I-Cas scalaire,
Ann. Fat. Sci. Toulouse 6, 215-255 (1984). 12. SERMANGE M.,
Mathematical
Opt. 14, 131-154(1986).
and numerical
aspects of one-dimensional
laminar
flame simulation,
Appl. Mufh.