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Other Methods in Time Discretization
Chapter 4 O t h e r M e t h o d s in T i m e D i s c r e t i z a t i o n
We have studied fully discrete finite element approximations for parabolic equations with the forward, the backward, and the Crank-Nicolson schemes for time discretization. However, many other methods have been examined in numerical computations. The present chapter is devoted to systematic studies for them.
4.1 Let
r(z)
Rational Approximation of Semigroups {e-tA}t>_o be a uniformly bounded (Co) semigroup on a Banach space X. We take
= (1 + z) -1 and describe the idea. It is a rational function provided with the properties
r(z) ~-- r
_.it_0 (]Zl p + I )
as z ~
0
for p = 1, and
Ir(z)l _< 1
for Rez >_ 0.
They are referred to as being of order p and A-acceptable, respectively. As we have seen, the operator r'~(TA) = (1 + TA) n is regarded as an approximation of e -tA for t = nT-, where ~- > 0 denotes the time mesh size. Based on those facts, we can argue as follows. First, the relation
d--~d (r~(sA)e_n(T_s)A)
=
nr n - l ( s A ) (r'(sA) -4- r ( s A ) ) Ae -n(r-s)A
-- nrn+l(sA)sA2e-n(~-s)A holds, and therefore we have [r~(TA) - e-nrA]A -2 - n
rn+l(sA)se -n(~--s)A ds.
Next, from the semigroup theory, the uniform boundedness of the semigroup represented as
lie-tAll _< C
(0 < t < oc)
(4.1)
is equivalent to the stability of its backward difference approximation:
]rn('rA)ll <_ C 145
(4.2)
4. Other Methods in Time Discretization
146 Those relations imply
II[rn(~-n) - e-n=A]A-P-lll <_ CtT p
(4.3)
for p = 1, where t = n7. Taking this observation in mind, we shall give error estimates for the approximate o p e r a t o r rn(rA) when the semigroup {e-tA}t>_o is holomorphic and r(z) is a general rational function of order p and A-acceptable. For this purpose, we recall the notion of A-acceptability of G. Dahlquist and generalize it in connection with a sector in the complex plane, where 0 E (0, 7r):
z0 = {z e C l 0 _< larg zl _< 0}. Namely, we introduce the following notations: 1~ T h e rational function
r(z) satisfies (i)o if Ir(z)l_ 1 for any z E E0 and Ir(oc)l < 1.
2 ~ T h e rational function
r(z) satisfies (ii)o if Ir(z)l < 1 for any z C E0 \ {0}.
3 ~ T h e rational function some 5 > 0.
r(z) satisfies (iii)o if Ir(z)l < 1 if 0 < Izl < ~ and z E E0 with
If I~(z)l _ 1 for any z E E0, then r(z) is said to be A0-acceptable. Therefore, Aacceptability means A~/2-acceptability in this terminology. Any rational function r(z) of order p (>_ 1) has p r o p e r t y (iii)o for 0 E (0, 7c/2). In fact, for p E [-0,0] we have 0
Op where ~ = x/~-l. Because of
r(O) = e-Zlz=o = 1
and
r'(0) =
r/
~-2-e-Z dz z=O
"- --1,
it holds t h a t c)---fi r(Ps
= - 2 R e e~ = - 2 c o s ~
_< - 2 c o s 0 < 0
(4.4)
p=0
and hence the conclusion follows. If r(z) is the Pad~ a p p r o x i m a t i o n of e -z with degrees n and m of the numerator and the d o m i n a t o r , respectively, then it is of order p = n + m. Furthermore, according to n < m, n = m, and n > m, it has properties (i)o, (ii)o, and (iii)o for some 0 E (0, 7r/2), respectively. In fact, in this case we have r(z) = R,,,m(z) = P,.m(z)/Q ......(z) with
P.,m(~)
(n + m - j)!n!
=
(~ + .~)!j!(~ _ j)! (-~-Y, j=O
Qnm(Z)
m
(n+m--j)!m!
x-'z_.,(,, + ,~,)b~(,~, - .J)~J j=O
4.1. Rational Approximation of Smigroups
147
The relation
]Rnm(Z)
-
I <_ c zl n+m+l
-
(z ~ 0)
is known. Let
(n + r e - j ) ! n ! anm(j) = (n + re)!j!(n-- j)! for 0 < j _< n. If n <_ re, we have
anm(j) > 0
and
a,~m(j) <_ anm(j).
j =0
T h e inequality
j =0
m
<-- E anm(J)PJ cosjqp j=o
= Re
Qnm(pe +zv) 5r 0
follows if m~p E [0, 7r/2]. Under the same assumption we also have n
IIm
Pnm(pe+'~)[ =
anm(j)(--p) j sin(-t-jqp)
_< E j =0
j =0 m
anm(J)PJ sinjqp
m
-< E a~m(J)PJ s i n j ~ = j =0
Q,~m(pe+~)l.
E anm(J)PJ sin(+jqD) = IIm j =o
Therefore, I~(p~)l ~ 1 holds for qp E [0, 7r/(2m)]. If n < re, r ( o c ) = 0 follows and (i)o holds with 0 = 7r/(2m). If n = re, we have Ir(oc)l = 1 < + o c . T h e n the inequality r(z)l < 1 for z E E~/2m is a consequence of the classical P h r a g m 6 n - L i n d e l 6 f theorem. The m a x i m u m principle now gives Ir(z)l < 1 for z E E0 \ {0} with 0 C (0, 7r/(2re)). This means t h a t r = Rnm satisfies (ii)o in this case. Finally, r(z) satisfies (iii)o always for some 0 E (0, 7r/2) as is described. It is known t h a t the Pad6 a p p r o x i m a t i o n Rnm of e -z is An/2-acceptable if and only if r e - 2 _< n _< re. If n = re and n = r e - 1, Rum is subject to a recursive formula based on the continued fraction expansion. Namely, we have the expansion e z = Z
1-
Z
1+
Z 2~
-
-
3+-.-
expressed by C z ---
Z
Z
Z
Z
Z
-- i+~-5+~-
Z
Z
g+-..+~-
2j_1
....
This implies eZ=l+
2Z
2-~
+
Z2
-6-
+
Z2
~
Z2
+ - - - + ~
2(2j-1)
+
9
4. Other Methods in Time Discretization
148
Accordingly, a rational approximation H2k+l(Z)= G2k+l(Z)/F2k+l(Z)of e z is introduced inductively with F1 = 1, Fa = 2 - z, G1 = 1, G3 = 2 + z, F2j+I
=
2 ( 2 j - 1)F~j_I + z2F2j_3,
and
Ggj+I = 2(2j - 1)G2j-1 + z2G2j-3. Similarly, H2k(z)= G2k(z)/F2k(z)is defined by Fo = 1, F2 = 1 - z, Go = 0, G2 = 1,
F2j =
{
2(2j-1)+
2 } 2j-3 z
2j-lz2F2j_4, F2j-2+2j 3
and
G2j = {2(2j - 1 ) +
2
2j- 3
"[
2j-
z f G2j-2 + 2j
1
3
G2j-4.
Then, the relations H 2 k + l ( - z ) = Rkk(z) and H 2 k ( - z ) = Rk-l,k(z) can be verified. Let A be an operator of type (00, M0) on a Banach space X, where 0o C (0, 7r/2) and Mo >_ 1. The holomorphic semigroup generated by - A is denoted by {e -tA }t>o. Suppose that A is bounded for simplicity. We can show the following. T h e o r e m 4.1. The estimate T
e-n'AII < C ( t )
P
(4.5)
holds for t = nT under one of the following assumptions, where fJ1 > 0 is a constant and 0>00" 1~ r(z) has property (i)o. 2 ~ r(z) has property (ii)o and 7 IIAI < M~ < +oo. 3 ~ r(z) has property (iii)o and T [JAIl < /l}I~ < ~.
We need several lemmas. constants, respectively.
Henceforth r > 0 and C > 0 stand for small and large
L e m m a 4.2. If a rational function r = r(z) is of order p and Ao-acceptable with p > 1 and 0 E (0, 7r/2), then there exist constants cy > 0 and fl > 0 such that [rn(z) - e -nz ] _< C n zl p+I e -''~'ee(z)
for z C Eo with
Izl
_ ~.
(4.6)
149
4.1. R a t i o n a l A p p r o x i m a t i o n o f Smigroups Proof." We have
rn(z)
--
e -nz ~-- s
r J - l ( z ) ( e - z -- r ( z ) )
e -(n-jT1)z
j=l
and
It(z)
-
e-Z[ <
C
Iz[ p+I
(Izl < ~o)
for some Oo > 0. This implies [r(z)[ _< e -Re(z) + C (Re(z)) p+I for z E E0 and Izl _< a0. T h e function f ( t ) = e - t + C t p+I satisfies f ( 0 ) = 1 and f'(0) < 0, and hence f ( t ) <_ e -~t holds on t E (0, al) with some /71 > 0 and ~ E (0, cos0). We get
for [zl <_ min(crl, a0) and z e E0. Therefore, if Izl _< (7 -- m i n ( a l , a0) and z e E0, we have
Irn(z) -- e -nz ] ~
s
e-(J-1)~Re(z)c [zip +1 e-(n-j+l)cosO'Re(z)
j=l C n ]Z[p+I e -/~nRetz) .
The proof is complete. Lemma
4.3.
Under the a s s u m p t i o n of Theorem ~.1, the estimate
II[<(~A) - e -~] A-~II ~ C < holds f o r t = h r . Proof: Take M > 0 satisfying r [IAII < M, ]~/'1 < M, and -s < M < 5 according to the cases in T h e o r e m 4.1, respectively. We may suppose t h a t a < M in Proposition 4.2. We take a p a t h of integration F0 divided into three parts, Fo = F1 U ['2 U ['3 with {90<01 <{9"
Pl
-
{)~=pg+ZOl [ O ~ p ~ g T - 1 } ,
r 2
=
{A =
r~ =
pc+Z~ I oT -1 < p < M T - 1 } ,
{M~-'~'~l
I~J < 0 1 }
Then, we have -
2m
=
I+II+III.
1
+
2
+
3
( r ~ ( r z ) - e-tz) z - p ( z I - A) -1 dz
4. Other Methods in Time Discretization
150 In use of Proposition 4.2, we have
IlZll <_ c =
c
n.Tp )P+ae-Znwc~176
/0
P
(.~-p?+'~-~(n'o)c~176 )
) - ~ d ~ -~ = C~-~. P
From the a s s u m p t i o n follows t h a t
I11:
__
c ]r
+
[ l ( z l - A)-'II dzl
2
C
fl_p~dp = CTP. T -1
P
Finally, the inequality
II(zI-
a)-l]l ~
7-
1
I z l - {IAII
M-
7-IIAll
holds on F3 and hence
IXIII ~_ c Jr ( ~(~z)l + 3
< c -
f
o,
0,
(M~-') -~
I - zl)Izl
-~' I ( A - A)-']I Idzl
Md~
M-~ItAII
-- CM Tp
follows. In the first two cases of the theorem, the constants C in the estimates of III]] and IIIIi] are independent of M, while CM goes to zero as M ---, +co. Therefore, the t h e o r e m follows. [] Proposition
4.4. I f r = r(z) satisfies (iii)o, then each 5' E (0, 5) admits/3 > 0 satisfying
Ir(z)l _< e -~'zj. If ~ = ~(z) satisfics (ii)o, thcn ~' > 0 can be taken arbitrarily large. Proof: As is noted, inequMity (4.4) holds. Any/30 E (0, cos0) admits a0 > 0 satisfying
I~(S~)I _< oxp (-/3op)
for p c (o, ~o) ~nd I~I-< 0. Next, let max {Ir(peW)llp E [c~0,~'], ~ C [-0,0]} = 1 - c < 1, and take /~1 > 0 satisfying exp (-/31~') = 1 - e .
Then, wc have
lr(pe'~)] _< exp (-/~,p) for p c (00, (Y) and [~p] < 0. S e t t i n g / 3 = min(/30, ~1), we get the conclusion.
[]
4.1. Rational Approximation of Smigroups
151
L e m m a 4.5. In the last two cases of Theorem ~.1, the estimate
IIA~<(wA)II < C~(n~) -~
(4.7)
holds for c~ > O. Proof: Taking a path of integration as in the proof of L e m m a 4.3, we have -
+ 27"~'~
=
1 -'~F2
(nTz)arn(Tz) (Z[ -- A) -1 dz 3
I+II.
From the assumption, we can take ~' of Proposition 4.4 in M < ~'. We have the following:
I111
<
IIHII
< -
Then
c
c
Z
(n"rz)C~e -~nrpc~
dD = Ca,
P
fie ( n T M ) ~ e - Z M n ~ d ~M< C ~ . ol
M-/1)1
-
the proof is complete.
[]
Helfrich's duality argument now gives the following.
Proof of Theorem ~.1 for the last two cases: 6, m E N a n d O _ < ~ - m _ < 1. We have
Divide N E N into n = t~ + m with
rn(TA) -- e-tA = (re(TA) -- e-e~A) A-Prm(~-A) + e-e~dA-p (r'~(TA) -- e-m~d) . Operator norm of the first term of the right-hand side is estimated as II[rl(TA) -- e-I~A]A-PlI . IIAPrm(TA)I < CT-P(mT) -p < C / n p = C(~-/t) p by Lemmas 4.3 and 4.5. T h a t of the second term is estimated similarly by the adjoint form of those lemmas. The proof is complete. [] Generally, r(c~) = 0 does not arise even in (i)o. We cannot take d' = +cx~ in Proposition 4.4. For the first case of Theorem 4.1 to prove, therefore, we require some more considerations. L e m m a 4.6. Let f l, f2 " R+ ~ R+ be continuous functions satisfying
n f l ( r ) dr < +oo r
and
7"
dr < +oo
for R > O, and let ~ be a meromorphic function provided with the properties IF(z)l < f~ (Jz[) and l~(z) - ~ ( o o ) l < f2(Izl) for lz[ <_ R and largz I = 0 ~ , where O~ > 0o. Then, the inequality
{ fl(r)~ dr + ~00Rf2(r) d~. + I~(oo)1
II~(A) < CR Jo(.R holds true.
4. Other Methods in Time Discretization
152 z
Proof." Letting h(z) = qp(z)- 1 + z qp(~176we have ~(A) = h ( A ) + ~(oo)A(1 + A) -1 Here, lid(1 + A ) - l l _ C holds and we have
[I~(A)II <_ IIh(A)ll + Cl~(c~)l. To estimate IIh(A)ll, we recall the path of integration F given in w oriented boundary of E01, which produce the equality
h(A) = ~
the positively
h(z) (zI - A) -1 dz.
We have z
lh(z)l = ~ ( z ) for Izl _< R and larg z I =
01,
I + ~ (oo)
f l ( [ Z [ ) - 4 - [ z [ [~(Cx:))l
and 1
Ih(z)l= ~(z)-~(~)+
1
l+z ~(~)
A(Iz[)+~l~(~)l
for Iz[ >__R and larg z I = 01, respectively. Those inequalities imply IIh(A)ll
_< c
=
C
{/0"
( k ( r ) + rl~(oc)l) -dr- +
{/0" 'r/0" fl(r)
-r- +
) } } -dr-
A ( r ) + -1I ~ ( r ) l
r
r
'" ( R +
f 2 ( r ) -r- +
~(oc)l
The proof is complete.
r
.
[]
Now, we complete the proof of Theorem 4.1.
Proof of Theorem 4.1 for the first case: Take 01 E (0o, 0) and a > 0 as in Proposition 4.2. The function ~(z) = e . . . . . r"(z) satisfies (4.6) if z E E0, and Iz <_ a. On the other hand, making ~ > 0 smaller if necessary, we may suppose that sup{Ir(z)ll
z~E0,,
zl _ ~ } = e
-z
from the assumption. Furthermore,
~(z)- r(o~)l_
cI ~-I
(Izl >_ ~)
holds because r = r(z) is rational. Those relation imply rt--1 j=O
4.2. Multi-step Method
153
and hence ~(~) - ~(oo)1 _< ~-~,zlcos0, +
c~-nZ/Izl
(4.8)
if z C E01 and Izl >_ a. Finally, we have
I~(oo)1 ~ e -n~.
(4.9)
Inequalities (4.6), (4.9), and (4.9) give
II~(~-A)II = II<(~-A)by L e m m a 4.6. The proof is complete. Theorem 4.1 is applicable to the semidiscrete finite element approximation of the parabolic equation, dub + Ahuh = 0 dt
(0 <_ t <_ T)
a~d
uh(O) = Phuo
in Xh with Ah, Xh, and Ph as before. We recall that the spectrum of Ah lies in a parabolic region in the complex plane uniformly in h. Each 0o E (0, rr/2) admits constants M0 _> 1 and ~ independent of h such that Ah -- .X is of type (00, Mo). Taking Vh = e-)'tUh instead of Uh, we can make the exponent 0 > 0 as small as we like in applying T h e o r e m 4.1. On the other hand, the inverse assumption guarantees the inequality IIAh[I <_ 7h -2 with some "y > 0. We have the following. T h e o r e m 4.7. Let r = r(z) be a rational function of order p (> 1) satisfying one of the following conditions for some 0 > O: 1~
(~)o.
2 ~ (ii)o and r h -2 < M1 < +c<~. 3 ~ (iii)o and r h -2 <_ M1 < "~-15. Then, the estimate (~.5) holds for t = n r with a constant C > 0 independent of h. Applied to the semidiscrete finite element approximation with higher accuracy, Theorem 4.1 gives natural results from the viewpoint of the correspondence of the rate of convergence with respect to the time discretization and the space discretization. Then, theorems on backward, forward and Crank-Nicolson schemes in w167 and 2.6 arise as special cases.
4.2
Multi-step
Method
Time discretization schemes studied in the preceding section may be called the singlestep m e t h o d totally, as the value un = u(tn) is determined by t h a t of un-1 = u(tn_l)
4. Other Methods in Time Discretization
154
for tn = nT. Multi-step m e t h o d of order q(>_ 2) determines Un from the values at the preceding q-steps, Un-1, U~-2,''" , U~_q, after determining u l , . . . , Uq-1 suitably. In this section we study this kind of schemes adopted to the evolution equation
du
dt
+Au=O
(O<_t<_T)
with
u(0)=uo
(4.10)
on a Banach space X, where A is of type (0, M) with 0 E (0, 7r/2) and M >_ 1, so that - A generates a holomorphic semigroup {e -tA}t_o. Suppose that A is bounded for simplicity. In the numerical scheme which we are studying, one determines U l , ' " , Uq-1 from u0 by a single-step approximation in use of a rational function first, and then computes Un+q through the relation q
E(ai
-t- TbiA)Itn+i : 0
(4.11)
i=0
for n = 0, 1 , - - . , where ai, bi E R. W i t h o u t loss of generality, we take aq = 1 in (4.11). Setting q
P(r
q
a~r
= E
and
S(() = E
i=0
bir
i=0
we call scheme (4.11) the multi-step m e t h o d (P, S). It is said to be of order p(_> 1) if the following relations hold:
Ea~=0, Eia,= Eb,,
q
q
q
EiJa~=iEiJ-lb~
q
q
i=0
i=0
i=0
i=1
/=1
(2
Letting 0 ~ = 1 and 0 . 0 -I = 0, we can write them simply as q
q
E ijai = j E iJ-lbi i=0
(4.12)
i=0
for j = 0, 1 , . . . ,p. To understand tile meaning of those equalities, take A = d/dx and compute tile Taylor expansion around r = 0 of the finite difference operator q
L~-[y] = E
[aiy(x + iT)+ fbiy'(x + iT)].
i=0
Then, equalities (4.12) follows if tile coefficients of T up to pth powers are let to be 0. If q = 1, scheme (4.11) is either one of the backward, the forward, the Crank-Nicolson, and the modified Crank-Nicolson (of order 1). Letting w((, z) = P ( ( ) + zS(() and 0 E (0, rr/2), we introduce the following notations: 1) The m e t h o d (P,S) satisfies (III)o if any root (j (1 _< j _< q) of P ( ( ) = 0 is simple and lies on the closed unit disk t(I _ 1, and moreover, if (j is in (j = 1 the inequality Re Aj/[Ajl > sin0 holds for Aj = CjS(Q)/P'(Q).
4.2. Multi-step Method
155
2) The m e t h o d (P, S) satisfies (II)o if it has property (III)o, and any root Cj(z) (1 _< j _< q) of w(r = 0 is simple and lies in the open unit disk [r < 1 for
z e x0 \ {0}.
3) The m e t h o d (P, S) satisfies (I)o if it has property (II)o, any root of S(~) = 0 is simple and lies in the open unit disk [C[ < 1, and bq > O. If (P, S) has property (III)o, then the requirements for Cy(Z) (1 < j < q) stated in (II)o holds if 0 < Iz] < ~ and z E E0, where ~ > 0 is a constant. In fact, Cj(z) is continuous in z and we may suppose t h a t Cj(0) -- Cj by re-ordering the number if necessary. We have only to show that [r
< 1 holds if 0 < [z I < ~ and
z e E0 for some n > 0, assuming I(jl = 1. Because ~j is simple, g(p, r differentiable in p at p = 0, where [r < 0: 09
8p
= [(j(pe~r
is
(o, r = 2Re (r162
The relation
r
- -s(r162
follows from P(Cj(z)) + zS(Cj(z)) = 0. We have ~p 0g (0 , Oh) = - 2 R e Aje ~r and hence sup
{ ~~1 6 2
r
holds. This gives the assertion. We make use of the rational function r(e) of order p - 1 to construct the approximate solution of (4.10) for the value of ui = u(ti) with i = 1 , - - . , q 1 and then take the multi-step m e t h o d (P, S) of order p for the values un = u(t,~) with n > q to determine. We have the following. Theorem
4.8. The inequality T
P
holds for t = nr ~ (0, T] with n > q, if one of the following condition is satisfied with 0>0o" 1) ( P , S ) and r have properties (I)o and (ii)o, respectively. 2) (P, S) and r have properties (III)o and (iii)o, respectively, together with the condi-
tion
IIAII _< Me <
min
6, ~, ~q[
.
Here, 6 > 0 is so taken as for r(z) to have no poles in z[ < ~ and z E Eo. Furthermore, ~ > 0 is the constant described above as (II)o holds for 0 < z I < ~ and z C Eo.
4. Other Methods in Time Discretization
156
In the second case, min(5, n, 1/lbql ) can be replaced by m i n ( ~ , l / b q [ ) and min((~,n), respectively, if (P, S) satisfies (II)o and bq > O, respectively. Setting
5~(z) = a~ + b~z aq + bqz
(1 _< i _< q),
we can write relation (4.11) as q
5,(TA)un+i = 0
(n = O, 1, 2 , . . . ) .
i=0
First, we study the functions un = u,~(z) of z E C satisfying q
E
5~(z)un+i(z) = 0
(4.13)
i=0
for n = 0, 1, 2 , . . . . We have the following. Lemma
4.9. If ~i(z) (1 <_ j <_ q) are distinct, then un(z) satisfies q
I~n(z)l <_ C ~ ICj(z)l" j=l
fo, ~ ,~ >_ q, ~h~,-~ c
>
o ~ a ~on, tant d ~ t ~ , ~ n ~ d
by
"~o,"" ,~q-,, ~Plr J
inf I C j ( z ) - r ~#j
Proof." From (4.13) follows q-1 i=0
while the relation q
p(r
+ zS(r
= (a~ + ~qZ) ~
(r - r
j=l
implies q-1
q
~ = - Z ~ ( z ) ~ ~ + 1-I (~ - ~ ( z ) ) i=O
j=l
with ~i = cri(z) (i = 0 , . . . , q - 1) determined by
{P(r162 (aq + bqz)
~-' i=0
and
4.2. Multi-step Method
157
Now we write ai and un for ~i(z) and un(z), respectively. If ui = (Q)i holds for i = 0, 1 , . . . , q - 1 with some j, then we have un = (Q)n for n = q, q + 1 , - . . In fact, we can show inductively that Un+q
~i'U'n+i
-- --
-- --
(~i ( ~ j ) i
. (O)n
--
(o)q+n
i=0
Therefore, in the case that q
(/=O,--.,q-1),
(4.14)
j:l
for some 0/j E C, we have q
(n=q,q+ 1,.-.). j=l
This implies q
<_ l<_j<_q m a x I jl
j=l
The linear transformation S"
(o/1,
..
. , 0/q)T~.+
..
(UO,
.
, U q _ 1 )T
is expressed by the matrix 9 9 9
~'1
"'"
1
~'q
9
~.-q-1
.
...
,
~'q--1
whose determinant is that of Vandermonde. Since Q's are distinct, UO,
" " " , ~q-1}
of (4.14) can represent an arbitrary element in C q. Then the l e m m a follows immediately. []
If some of Q(z) is not simple, the inequality
c(14-'/l)rn(z)--i
q
E j=l
ICj( )l
holds with m = re(z) > 2 standing for the m a x i m u m of their multiplicities.
4. Other Methods in Time Discretization
158
T u r n to the scheme in consideration. T h e a p p r o x i m a t i o n o p e r a t o r is denoted by T~(A):
un = T~ (A)uo Define the rational function sn(z) inductively as sn(z) = r(z) n for n = O, 1 , . - . , q - 1 and q
E
Si(Z)Sn+i(z) = 0
(n = O, 1, 2 , . . . ) .
j=o We have T , ~ ( A ) = s~(TA) and hence q 5
(z)
_
=
i=0 holds with q
Fj(z) = E 5~(z)e-(i+i)z"
(4.15)
i=o In the following lemma, the error o p e r a t o r T , : ( A ) - e -tA is represented by Fj('rA). Define the rational functions 7i(z) (j = 0, 4-1, 4 . 2 , . . . ) inductively as "~i(z) = 0 for j < 0, "),j(z) = 1 for j = 0, and q
=o k=O for j > 0. Lemma
4.10.
rite identity q
e -t"+~A - T~+q(A) = E 7 .... j('rA)Fj('rA) j=o q-1 j (4.16)
j=O k=0
holds for n = O, 1 , . . . Proof." E q u a l i t y (4.15) gives
j=o
j=o z=o n q j=o i=o n+q j=O
4.2. Multi-step Method
159
where J
Bj(z) -- ~
?,-k(z)(~j-k(Z)
(j : O, 1 , . . . , q)
k=O
and q
J~q+j(Z) -- E
"~/rt-J-k(Z)(~q-k(Z)
(j : O, 1, 2 , ' ' " ).
k=0
We have Bj(z) = 0 for j = q , . - . , n + q - 1 from the definition of 7j(z). We also have q
]~n+q(Z) ---
E ~/-k(Z)(~q-k(Z)
--- (~q(Z) = 1,
k=O
so that q--i
e--(n+q)z -- Sn+q(Z) --
~/n-J(Z)PJ(Z) -- E j=o
j
E
~/n-k(Z)(~J-k(Z) [e-Jz -- 3j(Z)]
j=0 k=o
follows. This implies the lemma.
[]
Estimate of
II~xp (-t~+~m) - T~+q(A)[[ is reduced to those of 7j(z) and Fj(z) in this way. For the former we have the following. 4.11. Let (P, S) have property (III)o, and suppose (II)o for 0 < Izl < and z c Eo with t~ > O. Then, each ~' e (O, min(~, 1/Ibql)) admits constants C > 0 and /3 > 0 satisfying
Proposition
for Izl < ~' and z E Eo. Proof." As we have seen, the roots Q(z) (1 _< j _< q) of P ( ( ) + inequalities
ICj(~)l _< ~-.,zj
z S ( ( ) = 0 satisfy the
(1~1 _< ~', z e r~0)
for some ~ > O. This implies
IO,n(Z)l < Ce -znlzl
(Izl < ~', z e Eo)
by L e m m a 4.9. Here, the constant C > 0 depends on % , - . . , ~[q-1, which are polynomials o f / ) 0 , " " , (~q-1. The latters are bounded on Izl _< ~' < 1/Ibql, hence C > 0 also is bounded there. Recall t h a t aq 1. The proof is complete. [] :
As for Fj(z) we have the following.
4. Other Methods in Time Discretization
160
P r o p o s i t i o n 4.12. /f (P,S) is of order p(>_ 1) and ec' < 1/Ibq] , then the inequality
IFj(z)l ~ C [zlp+I e-jRez holds for Re z >_ 0 and Iz <_ ~'. Proof: Letting v(t) = e -tz, we have F J ( z ) : ;-~"Si(z)e-(J+i)Z:(aq+bqz)-l
{ ~~:o - ~ " a i v ( j + i ) - ~ - ~i:o -"biv'(j+i)}
Here, (P, S) is of order p, and tile right-hand side is equal to
{~-~ fjj+l(J"~-i--t)Pv(P+l )(t) i=, a~ p!
(aq -t- bqz)-i
~-~lJ+i(J~-i--t)P-lv(p+l) ( p - 1)!
dt - i=, bi,y
In fact, we have
Aai -
fj+~
Jj
(j + p! i - t)Pv(P+l)(t ) dt
t=j+i
=
,,
g=O
(t)
t=j
-- -
-~.v(e)(j) + v(j + i) e=O
and
B~ =
jj
j+i (j + i -- t) p-1 )! v(V+l)(t) dt (p- 1
[ ~(j+i-t)e-lv(e)(t)]e=l ( f - 1),
t=j+i _- -
t=j
ie-1
_ ~
~v(e)(j)
+ v'(j + i).
e=l ( e - i)!
This implies q
q
i=1
i=0
-
e=l
C!
q
= E i=0
by {4.12}.
{aiv(j + i ) - bv'(j + i)}
i=0
i ea~ -
g
i=O
b~
dt
}
.
4.2. Multi-step Method In
use
161
of
_Fj(Z) = (aq -F bqz) -1
i=1
ai
p! JJ
•
-
with
v(p+l)(t) = (-z)p+le -tz,
we
(t) dt
i=1
bi
JJ
(p- 1
)!
v(P+l)(t) dt
}
get the desired inequality.
[]
Note that if bq > 0, we can take ec' < rc and a = oo in Propositions 4.11 and 4.12, respectively. Now, we can give the following.
Proof of Theorem ~.8 for the second case: Take A C
/1//1, rain
, n,
The rational function r(z) satisfies
I~j(z) - ~-Jz I < c I~" for 1 < j < q - l , z I < a, a n d z E Ee with s o m e a > 0. Recall (4.16). In the second term of the right-hand side, we have
I')'n-k(z)hj-k(z) [e -jk - rJ(z)] l <_ C Izl p e -z(~-k)lzt
(4.17)
for 0 < k _< j < q - 1, Izl < a0 = m i n ( a , ~ ' ) , and z E Ee. On the other hand, given M E (M1, ~'), we have
I'Yn-~(z)hj-~(z) [~-jz
~j(z)]l_<
C~,(~-~)lzl
forao <__ ] z ] _ < M a n d z E E o . We take a path of integration Fo = F1 U F2 U F3 with 01 E (0o, 0): pl
--
{Z---pe-t-zO,
I
0 ~ p ~ O-0T-1} ,
r~ =
{z = ~•176
F3 =
{A4T- l e zq~ I I~1 _< 01 }"
~o~-1< p_< M~-~},
It holds that
"/n-k(TA)(~J-k'(zA) [e-tjA-- rJ('rA)]
=
27c~
=
I+II+III.
1 (]; 1
+
fr 2
+
d / r3)
"Yn-k('rz)(~j-k.('rz)
[e -j~-z- rJ(Tz)] ( z I -
A) -1 dz
4. Other Methods in Time Discretization
162
Each term admits of the following estimates for n = q, q + 1,.--"
II111
c fo ~ ~-Z(n-k~'o(W)~-2 = C(ndp
<
k) -p < Cn -p
I
IZZll
<_ c
IlzIzll
< c
__ _ e-Z(n-k)wdP - c
o~--~
_pdp --<_Cn off(n-k) P e
P
-p,
e-Z(n-klMd~< Cn -p. 9
01
The second term of the right-hand side of (4.16) is estimated as
[q-1 j
Z Z ~,,_k(~-A),~j_,~(',-A)[~-,jA _ ,.j(.,_A)]
~ CTt -p.
j=0 k=0
To estimate the first term, we make use of Propositions 4.11 and 4.12. The inequality
Is j=o
<_Cne-~nlz',zP+l
follows for z I _ < M a n d z E E 0 .
In use of
j :0
1-It-I"2 =
-~-OfF ) s
(zZ-A) dz
3 j=0
-1
IV+V,
we get
II/Vll
<_ c
and
IlVll
< c -
ne_~nrP(7.p)p+l do = C r _ p P
~0 ~176
_01 .
0,
~.~-~""
M -
~
M1
dw < o~-~. -
The first term of the right-hand side of (4.16) is also estimated as
[["
~_ CTz -p.
j=O
The proof is complete. W h e n (P,S) has property (III)o, we can replace min(,~., 1/Ibql) by 1 / b q in Proposition 4.11, and accordingly, min(5, 1/[bql)by min(5, 1/bql ). Similarly, min(~, K, 1/Ibql)is replaced by min(~, K) in tile case of bq > 0 from tile reason described after the proof of Proposition 4.12.
4.2. Multi-step Method
163
Now we give the following.
Proof of theorem 4.8 for the first case:
Because (P,
ICj(z)l _< ~-z~zf
S) has
property
(I)o, the
inequality
( z l _ ~', = e zo)
holds for arbitrarily large ec' > 0 with s o m e / 3 > 0. T h e conclusion of Proposition 4.11 still holds for any ec' >> 1. M a k i n g / 3 > 0 smaller, we have
~j(z)l _< ~-~ < ,
( =1_ ~', = ~ x0),
I~'~(~)1 _< c~ -n~
(z >~', z~Eo)
(4.18)
so t h a t
follows as in the proof of Proposition 4.11. To estimate the first t e r m of the right-hand side of (4.16), we take M > r IIAII and represent n
1
j=0
2 7"i-~
E %_j(rA)Fy(rA) =
i nt_l-,2
+ ~r ) s %_j(rz)Fj(rz)(zI-A) 3
-1
dz
j=0
= I+II. Similarly to the case (b), we have
I-Zll _< o~-p with the constant C > 0 independent of M. If j _> 1, we get
Vn_j(~z)Fj(~z) (~Z- A) -1 & <_C
~-("-J)eM~+'~-jMc~176
3
01
of which the right-hand side goes to zero as M -+ oo. To estimate the term for j = 0, we make use of the a r g u m e n t s in the preceding section. First, we note lim Fo(z)= Izl~oo zEEo
bo bq
and b0
M a k i n g / 3 > 0 smaller if necessary, we have e-/3n
I-Tn(z)- "Tn(oo)l _< C ~
Iz
( el >_ R, z e r,o)
4. Other Methods in Time Discretization
164
for R > 0. Actually, from the proof of Lemma 4.9 this is reduced to e-~n
I~j(z)"- ~j(oo)nl ~ c ~
where ~j(z) (1 _< j _< q) denotes the root of P(r ~(z) o - ~(oo).l
:
(1=1 _> R, z ~ ~o),
Izl
+ zS(r
,~3(z) - ~ ( o o ) ,
(4.19)
= 0. Inequality (4.18) implies
n-1 . }2 ~;-~-~(z)~y(oo) k:0
<_ ~-"~ I~j(~) - ~ j ( ~ ) t , where
arises because (P, S) has property (I)o. Inequality (4.19) has been proven. On the other hand, If0(z)l _< c and
IF0(z)- F0(oo)l--
ao - bo/bq 1 +bqz
+ ~q
i=1
C -t- Ce -Izl cos01 7--7, Izl
~(z)~ -~
are obvious. We obtain b0 <_ C e - ~
%~(z)ro(~)- -~.(oO)Vqq
(1
)
~
+ e-tzl c~176 .
The first term of the right-hand side of (4.16) is estimated as
irj=0a by Lemma 4.6. We proceed to the second term. Recall (4.17) with
js ~ e_~,~Opp dp = Cn_ p P We note
+ I~._k(oc)~j_.(o~)] ]~J(z)- ~J(o~)].
(4.20)
Here, 0 _< k _< q - 1 in this term and inequality (4.19) assures e-~n
17,~-k(z)- 7n-~(oo)l < C ~
tzl
165
4.3. Product Formula for Izl ~ ~0 and z C E0. Furthermore, 3' and (~j-k satisfy C (~j-k(z)- (~j-k(oo)l _< 757 Izl
and
I t ( z ) - r(oc)l _<
C
Izl
for Izl _> a0, respectively, because they are rational functions. We also have 17n(CC)l < Ce -zn
and
I~n-k(z)l _< C.
Summing up those relations, we see that the right-hand side of (4.20) is dominated above by Ce-Zn/Izl. The second term of the right-hand side of (4.16) is estimated as q-1
j
E E 3`n-k(TA)(~J-k(TA)(e-t'A-- rJ(TA)) <_C/n".
j=0 k=0
The proof is complete.
4.3
Product
Formula
From the classical theory of Lie groups, it holds that one paraeter families of subgroups {exp ( - t A ) } t > o and {exp ( - t B ) } t > o satisfy
)in lim [r ( t ) ( o r n--,+c~ nt m
=exp(-t(A+B))
where r = exp ( - t A ) and r = exp ( - t B ) . H.F. Trotter extended this property to (Co) semigroups on Banach spaces. It has been called Trotter's product formula. Error estimates in the operator norm were known for bounded operators, but only strong convergence has been discussed for the other cases. However, D.L. Rogava has succeeded in giving them for analytic semigroups. We shall describe the simplest case. T h e o r e m 4.13. Let A1 and A2 be positive self-adjoint operators on a Hilbert space H satisfying IIAll/2e-tA2All/21 < C
(4.21)
for t e [0, J], and suppose that A = A1 + A2 is self-adjoint with D(A) = D(A1) N D(A2) and furthermore, D(Aal/2) C D(A~/2) N D ( A 3/2) holds. Then, T ( t ) = e-tA2e -tA1 satisfies the estimate
sup
tC[O,nJ]
IIe -tA -- T ( t ) n l l
with a constant C > 0 independent n = 1, 2, 3 , - - . .
< C---~ lx/'n
(422)
4. Other M e t h o d s in T i m e Discretization
166 This t h e o r e m bounded domain "10a = 0, and A2 We take a few
is applicable, for example, if A1 is the differential operator - A on a ft C R n with s m o o t h b o u n d a r y 0f~ provided with the b o u n d a r y condition = V ( z ) >_ 0 with V ( z ) sufficiently smooth. preliminaries. Let
A =
fO ~
AdE(A)
be the spectral decomposition of a positive self-adjoint A operator in H. t >_ 0 and c~ E [0, 2], we have
First, given
OG
[(I + t A ) -1 - e -tA] A -~ =
[(1 + tA) -1 - e -t~] (At) -~ d E ( A ) , t ~.
Because sups>0 I[(1 + s) -1 - e -~] s-~l < +oc, this implies
II [(I +
t A ) -1 - e-tA]
A- II
ct~.
(4.23)
(s > 0).
(4.24)
We also have tile elementary inequality (1 - e - S ) -'/2 <_ s -1/2 + 1
To see this, let f ( s ) = s - S / 2 - ( 1 - e - S ) -'/2. We have limst0 f ( s ) = + o c and l i m ~ + ~ f ( s ) = - 1 . To show i f ( s ) < 0 for s > 0, suppose the contrary. The case i f ( s ) = 0 is equivalent to
g(t) - t 2 - t -a - 31ogt = 0
fort=e
s/3 > 1.
However, this is impossible because g(1) = 0 and
g'(t) = t - 2 ( t -
1)(2t 2 + 2 t -
1) > 0
for t > 1.
Inequality (4.24) implies (1 - e - ~ ) -1 d lE(A)'u 2
=
<_
/0
<_
((tA) -1/2 + 1) 2d E ( A ) u 2 =
IIA- / ul
(t-1/2A-1/2 + I ) u l 2
+ I1~11)2
We obtain (4.25) forttCHandt>0. Now we show the following. w
167
4.3. Product Formula
L e m m a 4.14. The operator W ( t ) = e -tA - e-tA2e -tA1 satisfies
IIAT1/=W(t)A-~/=]I< Ct=
(4.26)
for t c=_[0, J]. Proof: We have
W ( t ) -- [e -tA - (I -it- tA) -1] -Jr-[(/-t-
t A ) -1 - ( I
+ tA2) -1 (I -t- tA1) -1]
+ [(I + tA2) -1 -- e -tA2] (I + tAl) -1 q- e -tA2 [(I + tA1) -1
-
-
e-tAil.
The first term of the right-hand side is estimated by (4.23) as
IIATX/2[e-tA-(I+tA)-l]A-a/2[[ IlA71/2AI/2[I II [e -tA - (I nt- ~A) -1] A-2I[ ~_~C~;2. Here, the inequality [IA1/UAT~/211 < C follows from Heinz' inequality and D(A~/2) C D(Aa/2). Similarly, the third and the fourth terms of the right-hand side are estimated as IlAT1/2[(I+tA2)-l-e-tA2](I+tA1)-lA-a/2[I
_<
IIAI~Ag'=[I II[(, + ~A~)-I - e-~=] A;=ll
and
IIAT1/2e-tA2[(I--l-t;A1)-l-e-tA1]A-3/211 IIAll/2e-tA2All/21t [[[(/-t-tA1) -1 - e -tA1] A1211 llAal/2A-3/211 ~ Ct 2, respectively. Finally, writing (I + tA) -1 - (I + tA2) -1 (I + tA1) -1 = (I + tA) -1. tA2 (1 + tA2) -1- tA1 (I + tA1) - t , we have
<_ I{ATI/=AI/]I I[(tA) The proof is complete.
(Z + tA)-lll [leA= (Z + tA=)-*ll lllllAT' A []
168
4. O t h e r M e t h o d s in T i m e Discretization
We are r e a d y to give the following. P r o o f of T h e o r e m 4.13: we p u t E(t)
-
e -tA
L e t t i n g t = n r , U ( n ) = e -nrA, U1 = e - r A ' , and U2 = e -rA2,
-
(
e -tn-lA2
)~ - -
. e -tn-lA]
g(n)
(82-gl)
-
n
and W = W(7-) = e -rA -- e - r A 2 . e -rA' T h e n , it holds t h a t
E(t)
-
8281. W . U(?z - 2) -Jv ...-Ji- (U2U1)n-1. W . U(O) 1) 4- U2U~/2. (I 81) 1/2" V. U(Tt 2) nt - ' ' " "Jr- U 2 U ~ / 2 . S n - 2 . ( I - 81) 1/2 V. U(0),
-
U1) -1/2
=
W U ( n - 1) +
=
WU(n
-
-
-
where V = (I
U~/2W
and
We o b t a i n
n-1 E(t)
=
W U ( n - 1) -I- U28~/2 ~
S k-1
(I
--
81) 1/9" V-U(Tt - 1~ -+- 1).
k=l Now we shall show
li~ 1(, ~,)1,~ I < ~
1~,,~
~4~/
for k = 1 , 2 , . . - . In fact, t a k i n g A / + 6 for A/ and m a k i n g 6 ~ 0, we may suppose t h a t A/ k 6, where i : 1, 2. In this case, we have
I1~' ~1)1'~1l< (' ~'~/1'~
'~,J <~"
~
'~'J ~ ~-'~
For s = e -ra C (0, 1), we o b t a i n
I1~~-' (, ~l)"'l < II~,ll~-1 Ill, f,~-' I1(1 ~')"~11 _~ 82(k-1)(1 Here, f ( O ) = / ( 1 )
-
-
8) 1/2 ~ 8k-1(1
= O, and m a x f (s) = f (So)
sE[O,1]
holds for So = (2k - 2 ) / ( 2 k - 1) C (0, 1). We get f ( s ) <_ (1 - So) 1/2 = ( 2 k -
1) -1/2
-
-
8) 1/2 ~ f(s).
169
C o m m e n t a r y to Chapter 4
and inequality (4.27) follows. On the other hand, we have U 1/2 (I - U1) -1/2 (TA1) 1/2 ~- ~o~176-z~- (1 -- e--TA) -1/2 (TA)I/2dE(A)
with SUps>o e -s/2 (1
-
e-S)-1/2 s 1/2 < +oc. This implies [IuI/2(I-U1)-I/2(TA1)I/2[]
<_C
and hence [IV. g ( n -
1)[[
k-
[IU1/2 ( Z - Ul) -1/2
(TA1)I/2II.T-1/2IAll/2wA-3/2II [[A3/2U(TL--k--1)[[
G C (n - k) -3/2
(4.28)
follows for k = 1, 2,--- , n - 1 if t E [0, nJ]. Inequalities (4.27) and (4.28)imply
I
2u 1
.
k=l <
.
.
. n-1
g2ll Igll1172- c . ~
i,i[ k - l / 2 ( n - k) -3/2
k=l 1-
9n-1 = C n -1/2
ft k=l because
1 ~l (k)-1/2 (
k) -3/2 ..~
n k=l
n
1
J0 1
x-1/2( 1 _ x)_a/2d z ,.0
()n _1
-1/2
The proof is complete.
Commentary
to C h a p t e r 4
4.1. This section follows the description of [148]. For later developments, see Crouzeix, Larsson, Piskarev, and Thom6e [100], Palencia [312], and Sava% [338]. See also the monograph Thom6e [383]. The notion of A-acceptability was introduced by Dahlquist [104], and studied by Ehle [118], axelsson [16], Cryer [103], Dahlquist [106], and others. For the equivalence between (4.1) and (4.2), see Yosida [410], Kato [205], and so forth. Recall that i f - A is a generator of a contraction semigroup, then constant C in those inequalities is taken to be 1.
4. Other Methods in Time Discretization
170
General rational approximations of (Co) semigroups were studied by Hersh and Kato [174] and Brenner and Thome~ [61]. Inequality (4.3) holds if r(z) is of order p and Aacceptable. In this section, we applied the method of Baker, Bramble, and Thom~e [25] for the approximation of holomorphic semigroups. A related work was done by Bramble and Thom~e [58]. The notions (i)o, (ii)o, and (iii)o were introduced by [148]. The Runge-Kutta method is regarded as a rational approximation of semigroups. Stability and error analysis were done by Crouzeix [97], Keeling [215], Ostermann and Roche [311], Lubich and Ostermann [254], [255], and Nakaguchi and Yagi [275] including the case applied for quasilinear equations. The fact that the Pad~ approximation r = R,.m of e -~ is An/2-acceptable if and only if m - 2 < n < rn was conjectured by Ehle and proven by Wanner, Hairer, and Norsett [402]. Related works are Birkoff and Varga [36], Ehle [119], and Saff and Varga [329]. Pad~ approximation Rnn(Z) can cast the implicit Runge-Kutta method of Butcher [69] applied to du/dt + Au = 0. The use of continued fraction expansion of the exponential function was proposed by Mori [270], where A-acceptability of Hn(z) is proven. It has also the practical use. Baker, Bramble, and Thom~e [25] showed Theorem 4.1 for the case that A is self-adjoint under weaker assumptions on r(z). The first case of this theorem is due to LeRoux [234]. Large constants in stability or error estimates can cause unreliabilities to the numerical computation. For this topic, see Lenferink and Spijker [232] and the references therein. 4.2. The first case of Theorem 4.8 was proven by LeRoux [234]. We have followed [148] for the notions (I)o, (If)o, and (IlI)o. Multi-step methods for ordinary and partial differential equations were also studied by Dahlquist [105], Zl~imal [421], Nevanlinnca [293], Raviart [324], Crouzeix [98], and Palencia [313]. 4.3. Product formula for (Co) semigroups was introduced by Trotter [388]. Nelson [292] gave a different proof making use of the Feyman path integral. Chernoff [78] extended the formula to a general principle as follows: Let {F(t)}t> 0 be a family of non-expansive operators and {e-tC}t>_ 0 a (Co) semigroup on a Banach space X. Suppose that )~ > 0 admits s-lira [I + At -1 ( I tl0
F(t,))] -1 :r = (I + )~C) - 1.'/;
for each :r E X. Then, it follows that s-limn_.~ F(t/n)'~z = e-tCz locally uniformly in t E [0, oo). Later it was generalized to nonlinear semigroups by Brezis and Pazy [64]. See w for details. On the other hand, Kato [206] studied unbounded perturbations, and Feit, Fleck, and Steiger [123], [124] applied the formula to find eigenvalues numerically. Related works were done by Chernoff [79], Kato and Masuda [208], and Kuroda and Toshio Suzuki
[228].
Concerning the case of bounded operators, see M. S~zuki [362], [363], [364] and the references therein. R.D. Rogava's paper is [327]. There, O ( l o g n / v ~ ) is asserted under the assumption of D(A1) C D(A2), instead of assuming (4.21) and D(A3~/2) C D(A32/2) A D(A3/2). Related works were done by Ichinose and Tamura [186], [187], [188].
We have studied fully discrete finite element approximations for parabolic equations with the forward, the backward, and the Crank-Nicolson schemes for time discretization. However, many other methods have been examined in numerical computations. The present chapter is devoted to systematic studies for them.
4.1 Let
r(z)
Rational Approximation of Semigroups {e-tA}t>_o be a uniformly bounded (Co) semigroup on a Banach space X. We take
= (1 + z) -1 and describe the idea. It is a rational function provided with the properties
r(z) ~-- r
_.it_0 (]Zl p + I )
as z ~
0
for p = 1, and
Ir(z)l _< 1
for Rez >_ 0.
They are referred to as being of order p and A-acceptable, respectively. As we have seen, the operator r'~(TA) = (1 + TA) n is regarded as an approximation of e -tA for t = nT-, where ~- > 0 denotes the time mesh size. Based on those facts, we can argue as follows. First, the relation
d--~d (r~(sA)e_n(T_s)A)
=
nr n - l ( s A ) (r'(sA) -4- r ( s A ) ) Ae -n(r-s)A
-- nrn+l(sA)sA2e-n(~-s)A holds, and therefore we have [r~(TA) - e-nrA]A -2 - n
rn+l(sA)se -n(~--s)A ds.
Next, from the semigroup theory, the uniform boundedness of the semigroup represented as
lie-tAll _< C
(0 < t < oc)
(4.1)
is equivalent to the stability of its backward difference approximation:
]rn('rA)ll <_ C 145
(4.2)
4. Other Methods in Time Discretization
146 Those relations imply
II[rn(~-n) - e-n=A]A-P-lll <_ CtT p
(4.3)
for p = 1, where t = n7. Taking this observation in mind, we shall give error estimates for the approximate o p e r a t o r rn(rA) when the semigroup {e-tA}t>_o is holomorphic and r(z) is a general rational function of order p and A-acceptable. For this purpose, we recall the notion of A-acceptability of G. Dahlquist and generalize it in connection with a sector in the complex plane, where 0 E (0, 7r):
z0 = {z e C l 0 _< larg zl _< 0}. Namely, we introduce the following notations: 1~ T h e rational function
r(z) satisfies (i)o if Ir(z)l_ 1 for any z E E0 and Ir(oc)l < 1.
2 ~ T h e rational function
r(z) satisfies (ii)o if Ir(z)l < 1 for any z C E0 \ {0}.
3 ~ T h e rational function some 5 > 0.
r(z) satisfies (iii)o if Ir(z)l < 1 if 0 < Izl < ~ and z E E0 with
If I~(z)l _ 1 for any z E E0, then r(z) is said to be A0-acceptable. Therefore, Aacceptability means A~/2-acceptability in this terminology. Any rational function r(z) of order p (>_ 1) has p r o p e r t y (iii)o for 0 E (0, 7c/2). In fact, for p E [-0,0] we have 0
Op where ~ = x/~-l. Because of
r(O) = e-Zlz=o = 1
and
r'(0) =
r/
~-2-e-Z dz z=O
"- --1,
it holds t h a t c)---fi r(Ps
= - 2 R e e~ = - 2 c o s ~
_< - 2 c o s 0 < 0
(4.4)
p=0
and hence the conclusion follows. If r(z) is the Pad~ a p p r o x i m a t i o n of e -z with degrees n and m of the numerator and the d o m i n a t o r , respectively, then it is of order p = n + m. Furthermore, according to n < m, n = m, and n > m, it has properties (i)o, (ii)o, and (iii)o for some 0 E (0, 7r/2), respectively. In fact, in this case we have r(z) = R,,,m(z) = P,.m(z)/Q ......(z) with
P.,m(~)
(n + m - j)!n!
=
(~ + .~)!j!(~ _ j)! (-~-Y, j=O
Qnm(Z)
m
(n+m--j)!m!
x-'z_.,(,, + ,~,)b~(,~, - .J)~J j=O
4.1. Rational Approximation of Smigroups
147
The relation
]Rnm(Z)
-
I <_ c zl n+m+l
-
(z ~ 0)
is known. Let
(n + r e - j ) ! n ! anm(j) = (n + re)!j!(n-- j)! for 0 < j _< n. If n <_ re, we have
anm(j) > 0
and
a,~m(j) <_ anm(j).
j =0
T h e inequality
j =0
m
<-- E anm(J)PJ cosjqp j=o
= Re
Qnm(pe +zv) 5r 0
follows if m~p E [0, 7r/2]. Under the same assumption we also have n
IIm
Pnm(pe+'~)[ =
anm(j)(--p) j sin(-t-jqp)
_< E j =0
j =0 m
anm(J)PJ sinjqp
m
-< E a~m(J)PJ s i n j ~ = j =0
Q,~m(pe+~)l.
E anm(J)PJ sin(+jqD) = IIm j =o
Therefore, I~(p~)l ~ 1 holds for qp E [0, 7r/(2m)]. If n < re, r ( o c ) = 0 follows and (i)o holds with 0 = 7r/(2m). If n = re, we have Ir(oc)l = 1 < + o c . T h e n the inequality r(z)l < 1 for z E E~/2m is a consequence of the classical P h r a g m 6 n - L i n d e l 6 f theorem. The m a x i m u m principle now gives Ir(z)l < 1 for z E E0 \ {0} with 0 C (0, 7r/(2re)). This means t h a t r = Rnm satisfies (ii)o in this case. Finally, r(z) satisfies (iii)o always for some 0 E (0, 7r/2) as is described. It is known t h a t the Pad6 a p p r o x i m a t i o n Rnm of e -z is An/2-acceptable if and only if r e - 2 _< n _< re. If n = re and n = r e - 1, Rum is subject to a recursive formula based on the continued fraction expansion. Namely, we have the expansion e z = Z
1-
Z
1+
Z 2~
-
-
3+-.-
expressed by C z ---
Z
Z
Z
Z
Z
-- i+~-5+~-
Z
Z
g+-..+~-
2j_1
....
This implies eZ=l+
2Z
2-~
+
Z2
-6-
+
Z2
~
Z2
+ - - - + ~
2(2j-1)
+
9
4. Other Methods in Time Discretization
148
Accordingly, a rational approximation H2k+l(Z)= G2k+l(Z)/F2k+l(Z)of e z is introduced inductively with F1 = 1, Fa = 2 - z, G1 = 1, G3 = 2 + z, F2j+I
=
2 ( 2 j - 1)F~j_I + z2F2j_3,
and
Ggj+I = 2(2j - 1)G2j-1 + z2G2j-3. Similarly, H2k(z)= G2k(z)/F2k(z)is defined by Fo = 1, F2 = 1 - z, Go = 0, G2 = 1,
F2j =
{
2(2j-1)+
2 } 2j-3 z
2j-lz2F2j_4, F2j-2+2j 3
and
G2j = {2(2j - 1 ) +
2
2j- 3
"[
2j-
z f G2j-2 + 2j
1
3
G2j-4.
Then, the relations H 2 k + l ( - z ) = Rkk(z) and H 2 k ( - z ) = Rk-l,k(z) can be verified. Let A be an operator of type (00, M0) on a Banach space X, where 0o C (0, 7r/2) and Mo >_ 1. The holomorphic semigroup generated by - A is denoted by {e -tA }t>o. Suppose that A is bounded for simplicity. We can show the following. T h e o r e m 4.1. The estimate T
e-n'AII < C ( t )
P
(4.5)
holds for t = nT under one of the following assumptions, where fJ1 > 0 is a constant and 0>00" 1~ r(z) has property (i)o. 2 ~ r(z) has property (ii)o and 7 IIAI < M~ < +oo. 3 ~ r(z) has property (iii)o and T [JAIl < /l}I~ < ~.
We need several lemmas. constants, respectively.
Henceforth r > 0 and C > 0 stand for small and large
L e m m a 4.2. If a rational function r = r(z) is of order p and Ao-acceptable with p > 1 and 0 E (0, 7r/2), then there exist constants cy > 0 and fl > 0 such that [rn(z) - e -nz ] _< C n zl p+I e -''~'ee(z)
for z C Eo with
Izl
_ ~.
(4.6)
149
4.1. R a t i o n a l A p p r o x i m a t i o n o f Smigroups Proof." We have
rn(z)
--
e -nz ~-- s
r J - l ( z ) ( e - z -- r ( z ) )
e -(n-jT1)z
j=l
and
It(z)
-
e-Z[ <
C
Iz[ p+I
(Izl < ~o)
for some Oo > 0. This implies [r(z)[ _< e -Re(z) + C (Re(z)) p+I for z E E0 and Izl _< a0. T h e function f ( t ) = e - t + C t p+I satisfies f ( 0 ) = 1 and f'(0) < 0, and hence f ( t ) <_ e -~t holds on t E (0, al) with some /71 > 0 and ~ E (0, cos0). We get
for [zl <_ min(crl, a0) and z e E0. Therefore, if Izl _< (7 -- m i n ( a l , a0) and z e E0, we have
Irn(z) -- e -nz ] ~
s
e-(J-1)~Re(z)c [zip +1 e-(n-j+l)cosO'Re(z)
j=l C n ]Z[p+I e -/~nRetz) .
The proof is complete. Lemma
4.3.
Under the a s s u m p t i o n of Theorem ~.1, the estimate
II[<(~A) - e -~] A-~II ~ C < holds f o r t = h r . Proof: Take M > 0 satisfying r [IAII < M, ]~/'1 < M, and -s < M < 5 according to the cases in T h e o r e m 4.1, respectively. We may suppose t h a t a < M in Proposition 4.2. We take a p a t h of integration F0 divided into three parts, Fo = F1 U ['2 U ['3 with {90<01 <{9"
Pl
-
{)~=pg+ZOl [ O ~ p ~ g T - 1 } ,
r 2
=
{A =
r~ =
pc+Z~ I oT -1 < p < M T - 1 } ,
{M~-'~'~l
I~J < 0 1 }
Then, we have -
2m
=
I+II+III.
1
+
2
+
3
( r ~ ( r z ) - e-tz) z - p ( z I - A) -1 dz
4. Other Methods in Time Discretization
150 In use of Proposition 4.2, we have
IlZll <_ c =
c
n.Tp )P+ae-Znwc~176
/0
P
(.~-p?+'~-~(n'o)c~176 )
) - ~ d ~ -~ = C~-~. P
From the a s s u m p t i o n follows t h a t
I11:
__
c ]r
+
[ l ( z l - A)-'II dzl
2
C
fl_p~dp = CTP. T -1
P
Finally, the inequality
II(zI-
a)-l]l ~
7-
1
I z l - {IAII
M-
7-IIAll
holds on F3 and hence
IXIII ~_ c Jr ( ~(~z)l + 3
< c -
f
o,
0,
(M~-') -~
I - zl)Izl
-~' I ( A - A)-']I Idzl
Md~
M-~ItAII
-- CM Tp
follows. In the first two cases of the theorem, the constants C in the estimates of III]] and IIIIi] are independent of M, while CM goes to zero as M ---, +co. Therefore, the t h e o r e m follows. [] Proposition
4.4. I f r = r(z) satisfies (iii)o, then each 5' E (0, 5) admits/3 > 0 satisfying
Ir(z)l _< e -~'zj. If ~ = ~(z) satisfics (ii)o, thcn ~' > 0 can be taken arbitrarily large. Proof: As is noted, inequMity (4.4) holds. Any/30 E (0, cos0) admits a0 > 0 satisfying
I~(S~)I _< oxp (-/3op)
for p c (o, ~o) ~nd I~I-< 0. Next, let max {Ir(peW)llp E [c~0,~'], ~ C [-0,0]} = 1 - c < 1, and take /~1 > 0 satisfying exp (-/31~') = 1 - e .
Then, wc have
lr(pe'~)] _< exp (-/~,p) for p c (00, (Y) and [~p] < 0. S e t t i n g / 3 = min(/30, ~1), we get the conclusion.
[]
4.1. Rational Approximation of Smigroups
151
L e m m a 4.5. In the last two cases of Theorem ~.1, the estimate
IIA~<(wA)II < C~(n~) -~
(4.7)
holds for c~ > O. Proof: Taking a path of integration as in the proof of L e m m a 4.3, we have -
+ 27"~'~
=
1 -'~F2
(nTz)arn(Tz) (Z[ -- A) -1 dz 3
I+II.
From the assumption, we can take ~' of Proposition 4.4 in M < ~'. We have the following:
I111
<
IIHII
< -
Then
c
c
Z
(n"rz)C~e -~nrpc~
dD = Ca,
P
fie ( n T M ) ~ e - Z M n ~ d ~M< C ~ . ol
M-/1)1
-
the proof is complete.
[]
Helfrich's duality argument now gives the following.
Proof of Theorem ~.1 for the last two cases: 6, m E N a n d O _ < ~ - m _ < 1. We have
Divide N E N into n = t~ + m with
rn(TA) -- e-tA = (re(TA) -- e-e~A) A-Prm(~-A) + e-e~dA-p (r'~(TA) -- e-m~d) . Operator norm of the first term of the right-hand side is estimated as II[rl(TA) -- e-I~A]A-PlI . IIAPrm(TA)I < CT-P(mT) -p < C / n p = C(~-/t) p by Lemmas 4.3 and 4.5. T h a t of the second term is estimated similarly by the adjoint form of those lemmas. The proof is complete. [] Generally, r(c~) = 0 does not arise even in (i)o. We cannot take d' = +cx~ in Proposition 4.4. For the first case of Theorem 4.1 to prove, therefore, we require some more considerations. L e m m a 4.6. Let f l, f2 " R+ ~ R+ be continuous functions satisfying
n f l ( r ) dr < +oo r
and
7"
dr < +oo
for R > O, and let ~ be a meromorphic function provided with the properties IF(z)l < f~ (Jz[) and l~(z) - ~ ( o o ) l < f2(Izl) for lz[ <_ R and largz I = 0 ~ , where O~ > 0o. Then, the inequality
{ fl(r)~ dr + ~00Rf2(r) d~. + I~(oo)1
II~(A) < CR Jo(.R holds true.
4. Other Methods in Time Discretization
152 z
Proof." Letting h(z) = qp(z)- 1 + z qp(~176we have ~(A) = h ( A ) + ~(oo)A(1 + A) -1 Here, lid(1 + A ) - l l _ C holds and we have
[I~(A)II <_ IIh(A)ll + Cl~(c~)l. To estimate IIh(A)ll, we recall the path of integration F given in w oriented boundary of E01, which produce the equality
h(A) = ~
the positively
h(z) (zI - A) -1 dz.
We have z
lh(z)l = ~ ( z ) for Izl _< R and larg z I =
01,
I + ~ (oo)
f l ( [ Z [ ) - 4 - [ z [ [~(Cx:))l
and 1
Ih(z)l= ~(z)-~(~)+
1
l+z ~(~)
A(Iz[)+~l~(~)l
for Iz[ >__R and larg z I = 01, respectively. Those inequalities imply IIh(A)ll
_< c
=
C
{/0"
( k ( r ) + rl~(oc)l) -dr- +
{/0" 'r/0" fl(r)
-r- +
) } } -dr-
A ( r ) + -1I ~ ( r ) l
r
r
'" ( R +
f 2 ( r ) -r- +
~(oc)l
The proof is complete.
r
.
[]
Now, we complete the proof of Theorem 4.1.
Proof of Theorem 4.1 for the first case: Take 01 E (0o, 0) and a > 0 as in Proposition 4.2. The function ~(z) = e . . . . . r"(z) satisfies (4.6) if z E E0, and Iz <_ a. On the other hand, making ~ > 0 smaller if necessary, we may suppose that sup{Ir(z)ll
z~E0,,
zl _ ~ } = e
-z
from the assumption. Furthermore,
~(z)- r(o~)l_
cI ~-I
(Izl >_ ~)
holds because r = r(z) is rational. Those relation imply rt--1 j=O
4.2. Multi-step Method
153
and hence ~(~) - ~(oo)1 _< ~-~,zlcos0, +
c~-nZ/Izl
(4.8)
if z C E01 and Izl >_ a. Finally, we have
I~(oo)1 ~ e -n~.
(4.9)
Inequalities (4.6), (4.9), and (4.9) give
II~(~-A)II = II<(~-A)by L e m m a 4.6. The proof is complete. Theorem 4.1 is applicable to the semidiscrete finite element approximation of the parabolic equation, dub + Ahuh = 0 dt
(0 <_ t <_ T)
a~d
uh(O) = Phuo
in Xh with Ah, Xh, and Ph as before. We recall that the spectrum of Ah lies in a parabolic region in the complex plane uniformly in h. Each 0o E (0, rr/2) admits constants M0 _> 1 and ~ independent of h such that Ah -- .X is of type (00, Mo). Taking Vh = e-)'tUh instead of Uh, we can make the exponent 0 > 0 as small as we like in applying T h e o r e m 4.1. On the other hand, the inverse assumption guarantees the inequality IIAh[I <_ 7h -2 with some "y > 0. We have the following. T h e o r e m 4.7. Let r = r(z) be a rational function of order p (> 1) satisfying one of the following conditions for some 0 > O: 1~
(~)o.
2 ~ (ii)o and r h -2 < M1 < +c<~. 3 ~ (iii)o and r h -2 <_ M1 < "~-15. Then, the estimate (~.5) holds for t = n r with a constant C > 0 independent of h. Applied to the semidiscrete finite element approximation with higher accuracy, Theorem 4.1 gives natural results from the viewpoint of the correspondence of the rate of convergence with respect to the time discretization and the space discretization. Then, theorems on backward, forward and Crank-Nicolson schemes in w167 and 2.6 arise as special cases.
4.2
Multi-step
Method
Time discretization schemes studied in the preceding section may be called the singlestep m e t h o d totally, as the value un = u(tn) is determined by t h a t of un-1 = u(tn_l)
4. Other Methods in Time Discretization
154
for tn = nT. Multi-step m e t h o d of order q(>_ 2) determines Un from the values at the preceding q-steps, Un-1, U~-2,''" , U~_q, after determining u l , . . . , Uq-1 suitably. In this section we study this kind of schemes adopted to the evolution equation
du
dt
+Au=O
(O<_t<_T)
with
u(0)=uo
(4.10)
on a Banach space X, where A is of type (0, M) with 0 E (0, 7r/2) and M >_ 1, so that - A generates a holomorphic semigroup {e -tA}t_o. Suppose that A is bounded for simplicity. In the numerical scheme which we are studying, one determines U l , ' " , Uq-1 from u0 by a single-step approximation in use of a rational function first, and then computes Un+q through the relation q
E(ai
-t- TbiA)Itn+i : 0
(4.11)
i=0
for n = 0, 1 , - - . , where ai, bi E R. W i t h o u t loss of generality, we take aq = 1 in (4.11). Setting q
P(r
q
a~r
= E
and
S(() = E
i=0
bir
i=0
we call scheme (4.11) the multi-step m e t h o d (P, S). It is said to be of order p(_> 1) if the following relations hold:
Ea~=0, Eia,= Eb,,
q
q
q
EiJa~=iEiJ-lb~
q
q
i=0
i=0
i=0
i=1
/=1
(2
Letting 0 ~ = 1 and 0 . 0 -I = 0, we can write them simply as q
q
E ijai = j E iJ-lbi i=0
(4.12)
i=0
for j = 0, 1 , . . . ,p. To understand tile meaning of those equalities, take A = d/dx and compute tile Taylor expansion around r = 0 of the finite difference operator q
L~-[y] = E
[aiy(x + iT)+ fbiy'(x + iT)].
i=0
Then, equalities (4.12) follows if tile coefficients of T up to pth powers are let to be 0. If q = 1, scheme (4.11) is either one of the backward, the forward, the Crank-Nicolson, and the modified Crank-Nicolson (of order 1). Letting w((, z) = P ( ( ) + zS(() and 0 E (0, rr/2), we introduce the following notations: 1) The m e t h o d (P,S) satisfies (III)o if any root (j (1 _< j _< q) of P ( ( ) = 0 is simple and lies on the closed unit disk t(I _ 1, and moreover, if (j is in (j = 1 the inequality Re Aj/[Ajl > sin0 holds for Aj = CjS(Q)/P'(Q).
4.2. Multi-step Method
155
2) The m e t h o d (P, S) satisfies (II)o if it has property (III)o, and any root Cj(z) (1 _< j _< q) of w(r = 0 is simple and lies in the open unit disk [r < 1 for
z e x0 \ {0}.
3) The m e t h o d (P, S) satisfies (I)o if it has property (II)o, any root of S(~) = 0 is simple and lies in the open unit disk [C[ < 1, and bq > O. If (P, S) has property (III)o, then the requirements for Cy(Z) (1 < j < q) stated in (II)o holds if 0 < Iz] < ~ and z E E0, where ~ > 0 is a constant. In fact, Cj(z) is continuous in z and we may suppose t h a t Cj(0) -- Cj by re-ordering the number if necessary. We have only to show that [r
< 1 holds if 0 < [z I < ~ and
z e E0 for some n > 0, assuming I(jl = 1. Because ~j is simple, g(p, r differentiable in p at p = 0, where [r < 0: 09
8p
= [(j(pe~r
is
(o, r = 2Re (r162
The relation
r
- -s(r162
follows from P(Cj(z)) + zS(Cj(z)) = 0. We have ~p 0g (0 , Oh) = - 2 R e Aje ~r and hence sup
{ ~~1 6 2
r
holds. This gives the assertion. We make use of the rational function r(e) of order p - 1 to construct the approximate solution of (4.10) for the value of ui = u(ti) with i = 1 , - - . , q 1 and then take the multi-step m e t h o d (P, S) of order p for the values un = u(t,~) with n > q to determine. We have the following. Theorem
4.8. The inequality T
P
holds for t = nr ~ (0, T] with n > q, if one of the following condition is satisfied with 0>0o" 1) ( P , S ) and r have properties (I)o and (ii)o, respectively. 2) (P, S) and r have properties (III)o and (iii)o, respectively, together with the condi-
tion
IIAII _< Me <
min
6, ~, ~q[
.
Here, 6 > 0 is so taken as for r(z) to have no poles in z[ < ~ and z E Eo. Furthermore, ~ > 0 is the constant described above as (II)o holds for 0 < z I < ~ and z C Eo.
4. Other Methods in Time Discretization
156
In the second case, min(5, n, 1/lbql ) can be replaced by m i n ( ~ , l / b q [ ) and min((~,n), respectively, if (P, S) satisfies (II)o and bq > O, respectively. Setting
5~(z) = a~ + b~z aq + bqz
(1 _< i _< q),
we can write relation (4.11) as q
5,(TA)un+i = 0
(n = O, 1, 2 , . . . ) .
i=0
First, we study the functions un = u,~(z) of z E C satisfying q
E
5~(z)un+i(z) = 0
(4.13)
i=0
for n = 0, 1, 2 , . . . . We have the following. Lemma
4.9. If ~i(z) (1 <_ j <_ q) are distinct, then un(z) satisfies q
I~n(z)l <_ C ~ ICj(z)l" j=l
fo, ~ ,~ >_ q, ~h~,-~ c
>
o ~ a ~on, tant d ~ t ~ , ~ n ~ d
by
"~o,"" ,~q-,, ~Plr J
inf I C j ( z ) - r ~#j
Proof." From (4.13) follows q-1 i=0
while the relation q
p(r
+ zS(r
= (a~ + ~qZ) ~
(r - r
j=l
implies q-1
q
~ = - Z ~ ( z ) ~ ~ + 1-I (~ - ~ ( z ) ) i=O
j=l
with ~i = cri(z) (i = 0 , . . . , q - 1) determined by
{P(r162 (aq + bqz)
~-' i=0
and
4.2. Multi-step Method
157
Now we write ai and un for ~i(z) and un(z), respectively. If ui = (Q)i holds for i = 0, 1 , . . . , q - 1 with some j, then we have un = (Q)n for n = q, q + 1 , - . . In fact, we can show inductively that Un+q
~i'U'n+i
-- --
-- --
(~i ( ~ j ) i
. (O)n
--
(o)q+n
i=0
Therefore, in the case that q
(/=O,--.,q-1),
(4.14)
j:l
for some 0/j E C, we have q
(n=q,q+ 1,.-.). j=l
This implies q
<_ l<_j<_q m a x I jl
j=l
The linear transformation S"
(o/1,
..
. , 0/q)T~.+
..
(UO,
.
, U q _ 1 )T
is expressed by the matrix 9 9 9
~'1
"'"
1
~'q
9
~.-q-1
.
...
,
~'q--1
whose determinant is that of Vandermonde. Since Q's are distinct, UO,
" " " , ~q-1}
of (4.14) can represent an arbitrary element in C q. Then the l e m m a follows immediately. []
If some of Q(z) is not simple, the inequality
c(14-'/l)rn(z)--i
q
E j=l
ICj( )l
holds with m = re(z) > 2 standing for the m a x i m u m of their multiplicities.
4. Other Methods in Time Discretization
158
T u r n to the scheme in consideration. T h e a p p r o x i m a t i o n o p e r a t o r is denoted by T~(A):
un = T~ (A)uo Define the rational function sn(z) inductively as sn(z) = r(z) n for n = O, 1 , . - . , q - 1 and q
E
Si(Z)Sn+i(z) = 0
(n = O, 1, 2 , . . . ) .
j=o We have T , ~ ( A ) = s~(TA) and hence q 5
(z)
_
=
i=0 holds with q
Fj(z) = E 5~(z)e-(i+i)z"
(4.15)
i=o In the following lemma, the error o p e r a t o r T , : ( A ) - e -tA is represented by Fj('rA). Define the rational functions 7i(z) (j = 0, 4-1, 4 . 2 , . . . ) inductively as "~i(z) = 0 for j < 0, "),j(z) = 1 for j = 0, and q
=o k=O for j > 0. Lemma
4.10.
rite identity q
e -t"+~A - T~+q(A) = E 7 .... j('rA)Fj('rA) j=o q-1 j (4.16)
j=O k=0
holds for n = O, 1 , . . . Proof." E q u a l i t y (4.15) gives
j=o
j=o z=o n q j=o i=o n+q j=O
4.2. Multi-step Method
159
where J
Bj(z) -- ~
?,-k(z)(~j-k(Z)
(j : O, 1 , . . . , q)
k=O
and q
J~q+j(Z) -- E
"~/rt-J-k(Z)(~q-k(Z)
(j : O, 1, 2 , ' ' " ).
k=0
We have Bj(z) = 0 for j = q , . - . , n + q - 1 from the definition of 7j(z). We also have q
]~n+q(Z) ---
E ~/-k(Z)(~q-k(Z)
--- (~q(Z) = 1,
k=O
so that q--i
e--(n+q)z -- Sn+q(Z) --
~/n-J(Z)PJ(Z) -- E j=o
j
E
~/n-k(Z)(~J-k(Z) [e-Jz -- 3j(Z)]
j=0 k=o
follows. This implies the lemma.
[]
Estimate of
II~xp (-t~+~m) - T~+q(A)[[ is reduced to those of 7j(z) and Fj(z) in this way. For the former we have the following. 4.11. Let (P, S) have property (III)o, and suppose (II)o for 0 < Izl < and z c Eo with t~ > O. Then, each ~' e (O, min(~, 1/Ibql)) admits constants C > 0 and /3 > 0 satisfying
Proposition
for Izl < ~' and z E Eo. Proof." As we have seen, the roots Q(z) (1 _< j _< q) of P ( ( ) + inequalities
ICj(~)l _< ~-.,zj
z S ( ( ) = 0 satisfy the
(1~1 _< ~', z e r~0)
for some ~ > O. This implies
IO,n(Z)l < Ce -znlzl
(Izl < ~', z e Eo)
by L e m m a 4.9. Here, the constant C > 0 depends on % , - . . , ~[q-1, which are polynomials o f / ) 0 , " " , (~q-1. The latters are bounded on Izl _< ~' < 1/Ibql, hence C > 0 also is bounded there. Recall t h a t aq 1. The proof is complete. [] :
As for Fj(z) we have the following.
4. Other Methods in Time Discretization
160
P r o p o s i t i o n 4.12. /f (P,S) is of order p(>_ 1) and ec' < 1/Ibq] , then the inequality
IFj(z)l ~ C [zlp+I e-jRez holds for Re z >_ 0 and Iz <_ ~'. Proof: Letting v(t) = e -tz, we have F J ( z ) : ;-~"Si(z)e-(J+i)Z:(aq+bqz)-l
{ ~~:o - ~ " a i v ( j + i ) - ~ - ~i:o -"biv'(j+i)}
Here, (P, S) is of order p, and tile right-hand side is equal to
{~-~ fjj+l(J"~-i--t)Pv(P+l )(t) i=, a~ p!
(aq -t- bqz)-i
~-~lJ+i(J~-i--t)P-lv(p+l) ( p - 1)!
dt - i=, bi,y
In fact, we have
Aai -
fj+~
Jj
(j + p! i - t)Pv(P+l)(t ) dt
t=j+i
=
,,
g=O
(t)
t=j
-- -
-~.v(e)(j) + v(j + i) e=O
and
B~ =
jj
j+i (j + i -- t) p-1 )! v(V+l)(t) dt (p- 1
[ ~(j+i-t)e-lv(e)(t)]e=l ( f - 1),
t=j+i _- -
t=j
ie-1
_ ~
~v(e)(j)
+ v'(j + i).
e=l ( e - i)!
This implies q
q
i=1
i=0
-
e=l
C!
q
= E i=0
by {4.12}.
{aiv(j + i ) - bv'(j + i)}
i=0
i ea~ -
g
i=O
b~
dt
}
.
4.2. Multi-step Method In
use
161
of
_Fj(Z) = (aq -F bqz) -1
i=1
ai
p! JJ
•
-
with
v(p+l)(t) = (-z)p+le -tz,
we
(t) dt
i=1
bi
JJ
(p- 1
)!
v(P+l)(t) dt
}
get the desired inequality.
[]
Note that if bq > 0, we can take ec' < rc and a = oo in Propositions 4.11 and 4.12, respectively. Now, we can give the following.
Proof of Theorem ~.8 for the second case: Take A C
/1//1, rain
, n,
The rational function r(z) satisfies
I~j(z) - ~-Jz I < c I~" for 1 < j < q - l , z I < a, a n d z E Ee with s o m e a > 0. Recall (4.16). In the second term of the right-hand side, we have
I')'n-k(z)hj-k(z) [e -jk - rJ(z)] l <_ C Izl p e -z(~-k)lzt
(4.17)
for 0 < k _< j < q - 1, Izl < a0 = m i n ( a , ~ ' ) , and z E Ee. On the other hand, given M E (M1, ~'), we have
I'Yn-~(z)hj-~(z) [~-jz
~j(z)]l_<
C~,(~-~)lzl
forao <__ ] z ] _ < M a n d z E E o . We take a path of integration Fo = F1 U F2 U F3 with 01 E (0o, 0): pl
--
{Z---pe-t-zO,
I
0 ~ p ~ O-0T-1} ,
r~ =
{z = ~•176
F3 =
{A4T- l e zq~ I I~1 _< 01 }"
~o~-1< p_< M~-~},
It holds that
"/n-k(TA)(~J-k'(zA) [e-tjA-- rJ('rA)]
=
27c~
=
I+II+III.
1 (]; 1
+
fr 2
+
d / r3)
"Yn-k('rz)(~j-k.('rz)
[e -j~-z- rJ(Tz)] ( z I -
A) -1 dz
4. Other Methods in Time Discretization
162
Each term admits of the following estimates for n = q, q + 1,.--"
II111
c fo ~ ~-Z(n-k~'o(W)~-2 = C(ndp
<
k) -p < Cn -p
I
IZZll
<_ c
IlzIzll
< c
__ _ e-Z(n-k)wdP - c
o~--~
_pdp --<_Cn off(n-k) P e
P
-p,
e-Z(n-klMd~< Cn -p. 9
01
The second term of the right-hand side of (4.16) is estimated as
[q-1 j
Z Z ~,,_k(~-A),~j_,~(',-A)[~-,jA _ ,.j(.,_A)]
~ CTt -p.
j=0 k=0
To estimate the first term, we make use of Propositions 4.11 and 4.12. The inequality
Is j=o
<_Cne-~nlz',zP+l
follows for z I _ < M a n d z E E 0 .
In use of
j :0
1-It-I"2 =
-~-OfF ) s
(zZ-A) dz
3 j=0
-1
IV+V,
we get
II/Vll
<_ c
and
IlVll
< c -
ne_~nrP(7.p)p+l do = C r _ p P
~0 ~176
_01 .
0,
~.~-~""
M -
~
M1
dw < o~-~. -
The first term of the right-hand side of (4.16) is also estimated as
[["
~_ CTz -p.
j=O
The proof is complete. W h e n (P,S) has property (III)o, we can replace min(,~., 1/Ibql) by 1 / b q in Proposition 4.11, and accordingly, min(5, 1/[bql)by min(5, 1/bql ). Similarly, min(~, K, 1/Ibql)is replaced by min(~, K) in tile case of bq > 0 from tile reason described after the proof of Proposition 4.12.
4.2. Multi-step Method
163
Now we give the following.
Proof of theorem 4.8 for the first case:
Because (P,
ICj(z)l _< ~-z~zf
S) has
property
(I)o, the
inequality
( z l _ ~', = e zo)
holds for arbitrarily large ec' > 0 with s o m e / 3 > 0. T h e conclusion of Proposition 4.11 still holds for any ec' >> 1. M a k i n g / 3 > 0 smaller, we have
~j(z)l _< ~-~ < ,
( =1_ ~', = ~ x0),
I~'~(~)1 _< c~ -n~
(z >~', z~Eo)
(4.18)
so t h a t
follows as in the proof of Proposition 4.11. To estimate the first t e r m of the right-hand side of (4.16), we take M > r IIAII and represent n
1
j=0
2 7"i-~
E %_j(rA)Fy(rA) =
i nt_l-,2
+ ~r ) s %_j(rz)Fj(rz)(zI-A) 3
-1
dz
j=0
= I+II. Similarly to the case (b), we have
I-Zll _< o~-p with the constant C > 0 independent of M. If j _> 1, we get
Vn_j(~z)Fj(~z) (~Z- A) -1 & <_C
~-("-J)eM~+'~-jMc~176
3
01
of which the right-hand side goes to zero as M -+ oo. To estimate the term for j = 0, we make use of the a r g u m e n t s in the preceding section. First, we note lim Fo(z)= Izl~oo zEEo
bo bq
and b0
M a k i n g / 3 > 0 smaller if necessary, we have e-/3n
I-Tn(z)- "Tn(oo)l _< C ~
Iz
( el >_ R, z e r,o)
4. Other Methods in Time Discretization
164
for R > 0. Actually, from the proof of Lemma 4.9 this is reduced to e-~n
I~j(z)"- ~j(oo)nl ~ c ~
where ~j(z) (1 _< j _< q) denotes the root of P(r ~(z) o - ~(oo).l
:
(1=1 _> R, z ~ ~o),
Izl
+ zS(r
,~3(z) - ~ ( o o ) ,
(4.19)
= 0. Inequality (4.18) implies
n-1 . }2 ~;-~-~(z)~y(oo) k:0
<_ ~-"~ I~j(~) - ~ j ( ~ ) t , where
arises because (P, S) has property (I)o. Inequality (4.19) has been proven. On the other hand, If0(z)l _< c and
IF0(z)- F0(oo)l--
ao - bo/bq 1 +bqz
+ ~q
i=1
C -t- Ce -Izl cos01 7--7, Izl
~(z)~ -~
are obvious. We obtain b0 <_ C e - ~
%~(z)ro(~)- -~.(oO)Vqq
(1
)
~
+ e-tzl c~176 .
The first term of the right-hand side of (4.16) is estimated as
irj=0a by Lemma 4.6. We proceed to the second term. Recall (4.17) with
js ~ e_~,~Opp dp = Cn_ p P We note
+ I~._k(oc)~j_.(o~)] ]~J(z)- ~J(o~)].
(4.20)
Here, 0 _< k _< q - 1 in this term and inequality (4.19) assures e-~n
17,~-k(z)- 7n-~(oo)l < C ~
tzl
165
4.3. Product Formula for Izl ~ ~0 and z C E0. Furthermore, 3' and (~j-k satisfy C (~j-k(z)- (~j-k(oo)l _< 757 Izl
and
I t ( z ) - r(oc)l _<
C
Izl
for Izl _> a0, respectively, because they are rational functions. We also have 17n(CC)l < Ce -zn
and
I~n-k(z)l _< C.
Summing up those relations, we see that the right-hand side of (4.20) is dominated above by Ce-Zn/Izl. The second term of the right-hand side of (4.16) is estimated as q-1
j
E E 3`n-k(TA)(~J-k(TA)(e-t'A-- rJ(TA)) <_C/n".
j=0 k=0
The proof is complete.
4.3
Product
Formula
From the classical theory of Lie groups, it holds that one paraeter families of subgroups {exp ( - t A ) } t > o and {exp ( - t B ) } t > o satisfy
)in lim [r ( t ) ( o r n--,+c~ nt m
=exp(-t(A+B))
where r = exp ( - t A ) and r = exp ( - t B ) . H.F. Trotter extended this property to (Co) semigroups on Banach spaces. It has been called Trotter's product formula. Error estimates in the operator norm were known for bounded operators, but only strong convergence has been discussed for the other cases. However, D.L. Rogava has succeeded in giving them for analytic semigroups. We shall describe the simplest case. T h e o r e m 4.13. Let A1 and A2 be positive self-adjoint operators on a Hilbert space H satisfying IIAll/2e-tA2All/21 < C
(4.21)
for t e [0, J], and suppose that A = A1 + A2 is self-adjoint with D(A) = D(A1) N D(A2) and furthermore, D(Aal/2) C D(A~/2) N D ( A 3/2) holds. Then, T ( t ) = e-tA2e -tA1 satisfies the estimate
sup
tC[O,nJ]
IIe -tA -- T ( t ) n l l
with a constant C > 0 independent n = 1, 2, 3 , - - . .
< C---~ lx/'n
(422)
4. Other M e t h o d s in T i m e Discretization
166 This t h e o r e m bounded domain "10a = 0, and A2 We take a few
is applicable, for example, if A1 is the differential operator - A on a ft C R n with s m o o t h b o u n d a r y 0f~ provided with the b o u n d a r y condition = V ( z ) >_ 0 with V ( z ) sufficiently smooth. preliminaries. Let
A =
fO ~
AdE(A)
be the spectral decomposition of a positive self-adjoint A operator in H. t >_ 0 and c~ E [0, 2], we have
First, given
OG
[(I + t A ) -1 - e -tA] A -~ =
[(1 + tA) -1 - e -t~] (At) -~ d E ( A ) , t ~.
Because sups>0 I[(1 + s) -1 - e -~] s-~l < +oc, this implies
II [(I +
t A ) -1 - e-tA]
A- II
ct~.
(4.23)
(s > 0).
(4.24)
We also have tile elementary inequality (1 - e - S ) -'/2 <_ s -1/2 + 1
To see this, let f ( s ) = s - S / 2 - ( 1 - e - S ) -'/2. We have limst0 f ( s ) = + o c and l i m ~ + ~ f ( s ) = - 1 . To show i f ( s ) < 0 for s > 0, suppose the contrary. The case i f ( s ) = 0 is equivalent to
g(t) - t 2 - t -a - 31ogt = 0
fort=e
s/3 > 1.
However, this is impossible because g(1) = 0 and
g'(t) = t - 2 ( t -
1)(2t 2 + 2 t -
1) > 0
for t > 1.
Inequality (4.24) implies (1 - e - ~ ) -1 d lE(A)'u 2
=
<_
/0
<_
((tA) -1/2 + 1) 2d E ( A ) u 2 =
IIA- / ul
(t-1/2A-1/2 + I ) u l 2
+ I1~11)2
We obtain (4.25) forttCHandt>0. Now we show the following. w
167
4.3. Product Formula
L e m m a 4.14. The operator W ( t ) = e -tA - e-tA2e -tA1 satisfies
IIAT1/=W(t)A-~/=]I< Ct=
(4.26)
for t c=_[0, J]. Proof: We have
W ( t ) -- [e -tA - (I -it- tA) -1] -Jr-[(/-t-
t A ) -1 - ( I
+ tA2) -1 (I -t- tA1) -1]
+ [(I + tA2) -1 -- e -tA2] (I + tAl) -1 q- e -tA2 [(I + tA1) -1
-
-
e-tAil.
The first term of the right-hand side is estimated by (4.23) as
IIATX/2[e-tA-(I+tA)-l]A-a/2[[ IlA71/2AI/2[I II [e -tA - (I nt- ~A) -1] A-2I[ ~_~C~;2. Here, the inequality [IA1/UAT~/211 < C follows from Heinz' inequality and D(A~/2) C D(Aa/2). Similarly, the third and the fourth terms of the right-hand side are estimated as IlAT1/2[(I+tA2)-l-e-tA2](I+tA1)-lA-a/2[I
_<
IIAI~Ag'=[I II[(, + ~A~)-I - e-~=] A;=ll
and
IIAT1/2e-tA2[(I--l-t;A1)-l-e-tA1]A-3/211 IIAll/2e-tA2All/21t [[[(/-t-tA1) -1 - e -tA1] A1211 llAal/2A-3/211 ~ Ct 2, respectively. Finally, writing (I + tA) -1 - (I + tA2) -1 (I + tA1) -1 = (I + tA) -1. tA2 (1 + tA2) -1- tA1 (I + tA1) - t , we have
<_ I{ATI/=AI/]I I[(tA) The proof is complete.
(Z + tA)-lll [leA= (Z + tA=)-*ll lllllAT' A []
168
4. O t h e r M e t h o d s in T i m e Discretization
We are r e a d y to give the following. P r o o f of T h e o r e m 4.13: we p u t E(t)
-
e -tA
L e t t i n g t = n r , U ( n ) = e -nrA, U1 = e - r A ' , and U2 = e -rA2,
-
(
e -tn-lA2
)~ - -
. e -tn-lA]
g(n)
(82-gl)
-
n
and W = W(7-) = e -rA -- e - r A 2 . e -rA' T h e n , it holds t h a t
E(t)
-
8281. W . U(?z - 2) -Jv ...-Ji- (U2U1)n-1. W . U(O) 1) 4- U2U~/2. (I 81) 1/2" V. U(Tt 2) nt - ' ' " "Jr- U 2 U ~ / 2 . S n - 2 . ( I - 81) 1/2 V. U(0),
-
U1) -1/2
=
W U ( n - 1) +
=
WU(n
-
-
-
where V = (I
U~/2W
and
We o b t a i n
n-1 E(t)
=
W U ( n - 1) -I- U28~/2 ~
S k-1
(I
--
81) 1/9" V-U(Tt - 1~ -+- 1).
k=l Now we shall show
li~ 1(, ~,)1,~ I < ~
1~,,~
~4~/
for k = 1 , 2 , . . - . In fact, t a k i n g A / + 6 for A/ and m a k i n g 6 ~ 0, we may suppose t h a t A/ k 6, where i : 1, 2. In this case, we have
I1~' ~1)1'~1l< (' ~'~/1'~
'~,J <~"
~
'~'J ~ ~-'~
For s = e -ra C (0, 1), we o b t a i n
I1~~-' (, ~l)"'l < II~,ll~-1 Ill, f,~-' I1(1 ~')"~11 _~ 82(k-1)(1 Here, f ( O ) = / ( 1 )
-
-
8) 1/2 ~ 8k-1(1
= O, and m a x f (s) = f (So)
sE[O,1]
holds for So = (2k - 2 ) / ( 2 k - 1) C (0, 1). We get f ( s ) <_ (1 - So) 1/2 = ( 2 k -
1) -1/2
-
-
8) 1/2 ~ f(s).
169
C o m m e n t a r y to Chapter 4
and inequality (4.27) follows. On the other hand, we have U 1/2 (I - U1) -1/2 (TA1) 1/2 ~- ~o~176-z~- (1 -- e--TA) -1/2 (TA)I/2dE(A)
with SUps>o e -s/2 (1
-
e-S)-1/2 s 1/2 < +oc. This implies [IuI/2(I-U1)-I/2(TA1)I/2[]
<_C
and hence [IV. g ( n -
1)[[
k-
[IU1/2 ( Z - Ul) -1/2
(TA1)I/2II.T-1/2IAll/2wA-3/2II [[A3/2U(TL--k--1)[[
G C (n - k) -3/2
(4.28)
follows for k = 1, 2,--- , n - 1 if t E [0, nJ]. Inequalities (4.27) and (4.28)imply
I
2u 1
.
k=l <
.
.
. n-1
g2ll Igll1172- c . ~
i,i[ k - l / 2 ( n - k) -3/2
k=l 1-
9n-1 = C n -1/2
ft k=l because
1 ~l (k)-1/2 (
k) -3/2 ..~
n k=l
n
1
J0 1
x-1/2( 1 _ x)_a/2d z ,.0
()n _1
-1/2
The proof is complete.
Commentary
to C h a p t e r 4
4.1. This section follows the description of [148]. For later developments, see Crouzeix, Larsson, Piskarev, and Thom6e [100], Palencia [312], and Sava% [338]. See also the monograph Thom6e [383]. The notion of A-acceptability was introduced by Dahlquist [104], and studied by Ehle [118], axelsson [16], Cryer [103], Dahlquist [106], and others. For the equivalence between (4.1) and (4.2), see Yosida [410], Kato [205], and so forth. Recall that i f - A is a generator of a contraction semigroup, then constant C in those inequalities is taken to be 1.
4. Other Methods in Time Discretization
170
General rational approximations of (Co) semigroups were studied by Hersh and Kato [174] and Brenner and Thome~ [61]. Inequality (4.3) holds if r(z) is of order p and Aacceptable. In this section, we applied the method of Baker, Bramble, and Thom~e [25] for the approximation of holomorphic semigroups. A related work was done by Bramble and Thom~e [58]. The notions (i)o, (ii)o, and (iii)o were introduced by [148]. The Runge-Kutta method is regarded as a rational approximation of semigroups. Stability and error analysis were done by Crouzeix [97], Keeling [215], Ostermann and Roche [311], Lubich and Ostermann [254], [255], and Nakaguchi and Yagi [275] including the case applied for quasilinear equations. The fact that the Pad~ approximation r = R,.m of e -~ is An/2-acceptable if and only if m - 2 < n < rn was conjectured by Ehle and proven by Wanner, Hairer, and Norsett [402]. Related works are Birkoff and Varga [36], Ehle [119], and Saff and Varga [329]. Pad~ approximation Rnn(Z) can cast the implicit Runge-Kutta method of Butcher [69] applied to du/dt + Au = 0. The use of continued fraction expansion of the exponential function was proposed by Mori [270], where A-acceptability of Hn(z) is proven. It has also the practical use. Baker, Bramble, and Thom~e [25] showed Theorem 4.1 for the case that A is self-adjoint under weaker assumptions on r(z). The first case of this theorem is due to LeRoux [234]. Large constants in stability or error estimates can cause unreliabilities to the numerical computation. For this topic, see Lenferink and Spijker [232] and the references therein. 4.2. The first case of Theorem 4.8 was proven by LeRoux [234]. We have followed [148] for the notions (I)o, (If)o, and (IlI)o. Multi-step methods for ordinary and partial differential equations were also studied by Dahlquist [105], Zl~imal [421], Nevanlinnca [293], Raviart [324], Crouzeix [98], and Palencia [313]. 4.3. Product formula for (Co) semigroups was introduced by Trotter [388]. Nelson [292] gave a different proof making use of the Feyman path integral. Chernoff [78] extended the formula to a general principle as follows: Let {F(t)}t> 0 be a family of non-expansive operators and {e-tC}t>_ 0 a (Co) semigroup on a Banach space X. Suppose that )~ > 0 admits s-lira [I + At -1 ( I tl0
F(t,))] -1 :r = (I + )~C) - 1.'/;
for each :r E X. Then, it follows that s-limn_.~ F(t/n)'~z = e-tCz locally uniformly in t E [0, oo). Later it was generalized to nonlinear semigroups by Brezis and Pazy [64]. See w for details. On the other hand, Kato [206] studied unbounded perturbations, and Feit, Fleck, and Steiger [123], [124] applied the formula to find eigenvalues numerically. Related works were done by Chernoff [79], Kato and Masuda [208], and Kuroda and Toshio Suzuki
[228].
Concerning the case of bounded operators, see M. S~zuki [362], [363], [364] and the references therein. R.D. Rogava's paper is [327]. There, O ( l o g n / v ~ ) is asserted under the assumption of D(A1) C D(A2), instead of assuming (4.21) and D(A3~/2) C D(A32/2) A D(A3/2). Related works were done by Ichinose and Tamura [186], [187], [188].