ETNA
Electronic Transactions on Numerical Analysis.
Volume 30, pp. 237-246, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
ASYMPTOTIC BEHAVIOR FOR NUMERICAL SOLUTIONS OF A SEMILINEAR
PARABOLIC EQUATION WITH A NONLINEAR BOUNDARY CONDITION
NABONGO DIABATE AND THÉODORE K. BONI
Abstract. This paper concerns the study of the numerical approximation for the following initial-boundary
value problem,
!"
#$"%'&("#'
%'&)*+,-#.
.'&" /"
0#$1.&234"#$& 5/"67/879
where : , *;3 and < =>? . We show that the solution of a semidiscrete form of the initial value problem
above goes to zero as approaches infinity and give its asymptotic behavior. We provide some numerical experiments
that illustrate our analysis.
Key words. semidiscretizations, semilinear parabolic equation, asymptotic behavior, convergence
AMS subject classifications. 35B40, 35B50, 35K60, 65M06
1. Introduction. Consider the following initial-boundary value problem,
(1.1)
(1.2)
(1.3)
@ACBD@E"EGFIHJ@(K1LNMPORQSOUTLWVXRMYL
@ E(Z'M(L[V[\]BDMYLW@ EYZ^TL[V[\`_bac@ deZ^TLfV[\]BDM(LgVXhMYL
@CZiQjLkM\]BD@2lJZiQ \>mbM(LNMnRQSnoTL
d
where H/XhM , a:XRM , p XhqrXsT , @ lt9uwv Z[x MYL-Tyi\ , @ lz Z.M\]BUM , and @ lz Z[T"\`_bac@ l Z^T\{B|M .
The theoretical study of asymptotic behavior of solutions for semilinear parabolic equations has been the subject of investigations of many authors; see [2, 4] and the references
cited therein. In particular in [4], when aGB}M , the authors have shown that the solution @ of
(1.1)–(1.3) decays to zero as V goes to infinity and satisfies
~€$
A'‚wƒ V…†‡ „ „Yˆ @‰Z'QjLfV[\ ˆ ƒB
u
BZ Š[‹ v
\ †‡ „ „ . Similar results have been obtained in [2] in the case where awXDM
where u
KŒ v[Ž
and p XqhXT . Indeed, in this case, it is shown that the solution @ decays to zero as V
~€$ A'‚wƒbV‘’^‡ „ @‰Z'QjL[V[\ ƒB u l
approaches infinity and
where u l8B“Z Š[‹ d v Œ v[Ž \…’”‡ „ „ .
„(ˆ
ˆ
In this paper we are interesting in the numerical study of (1.1)–(1.3). First, using a
semidiscrete form of (1.1)–(1.3), we prove similar results for the semidiscrete solution. Previously, authors have used numerical methods to study the phenomenon of blow-up and the
one of extinction; see [1, 3]. This paper is organized as follows. In the Section 2, we give
a semidiscrete form of (1.1)–(1.3) and we prove some results about the discrete maximum
principle. In Section 3, we show that the semidiscrete solution goes to zeros as V goes to
infinity and give its asymptotic behavior. In Section 4, we prove that the semidiscrete scheme
converges. Finally, in Section 5, we give some numerical results.
•
Received November 26, 2007. Accepted for publication May 22, 2008. Published online on September 12,
2008.
Recommended by O. Widlund.
Université d’Abobo-Adjamé, UFR-SFA, Département de Mathématiques et Informatiques, 16 BP 372 Abidjan 16, Cote d’Ivoire (nabongo diabate@yahoo.fr).
Institut National Polytechnique Houphouet-Boigny de Yamoussoukro, BP 1093 Yamoussoukro, Cote d’Ivoire
(theokboni@yahoo.fr).
237
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238
N. DIABATE AND T. K. BONI
2. The semidiscrete scheme. In this section, we give some lemmas which will be
used later. Let – be a positive integer, and define the grid Q˜—!Bš™^› , Msnœ™?n – , where
›BTž – . We approximate the solution @ of the problem (1.1)–(1.3) by the solution
Ÿ] ZiV[\]B“Z Ÿ l Z'V[\
L Ÿ Z'V[\
L-¡¢¡¢¡-L Ÿ]£ ZiV[\f\^¤
of the semidiscrete equations,
v
¥ Ÿ —kZiV[\
BD¦§ Ÿ — Z'V[\CFIH Z Ÿ — Z'V[\[\ K LgM¨nR™]n – FRTLWVXRM(L
¥ V
(2.1)
¥ Ÿ £ ZiV[\
a
BD¦ § Ÿ £ Z'V[\CFIH Z Ÿ £ ZiV[\[\+KFª© Z Ÿ £ ZiV[\f\ d L«V>XRMYL
¥ V
(2.2)
›
Ÿ — Z.M\{B Ÿ — l mhMYLWM¨nb™{n
(2.3)
– L
where
¦ § Ÿ — Z'V[\]B
¦§ Ÿ l Z'V[\]B
Ÿ —€¬
Ÿ — Z'V[\j_ Ÿ — 'Z V[\
Πv
L­TnR™{n – FbTL
› §
Ÿ Z'V[\CF Ÿ lJZiV[\
Ÿ £
Z'V[\CF Ÿ £ ZiV[\
Πv
© v
©
©
LN¦§ Ÿ;£ iZ V[\{B ©
¡
› §
› §
v
ZiV[\CF
©
The following lemma is a semidiscrete
form£ ¬ of the maximum principle.
£ ¬
l
v \ and let ± Z'V[\ t9u v Zfx M(L[®²y+Lf°
v \ such that
L EMMA 2.1. Let H ZiV[\ t9u Z[x MYL[®¯y%Lk°
¥
± — Z'V[\ F‘¦ § — Z'V[\j_³H — ZiV[\ — ZiV[\mhMYLWM¨nR™]n
¥ V
±
±
± — Z'M\mhMYLWM¨nR™]n
– LWV t Z.MYLf®8\
L
– ¡
Then we have ± — ZiV[\>mhM for MnR™]n – , V t Z'M(L[®8\ .
¨$µ l-¶—'¶ £-· l-¶2A+¶
—
Proof. Let ® l O“® and let ´ B
¤ ¸ ± ZiV[\ . Since, ± 'Z V[\ is a continuous
function on the compact x MYLf® l y , there exists V l¨t x MYLf® l y such that ´ B ± — ¸ Z'V l \ for a certain
™ l¹t?º MYL-¡¢¡¢¡¢L –(» . It is not hard to see that
¥
~$$ ± — ¸ ZiV”l\CF ± — ¸ Z'V”l¯F…½Y\ nRM(L
¼ ‚l
½
i
Z
”
V

l
C
\
F
l
ZiV”l\
¦ § ± — ¸ ZiV”l\]B¾© ± v
© ±
mbM €¿ ™+l8B|M(L
(2.5)
› §
— ¬ ZiV l \CF ± — ZiV l \À_ ± —
l
¸ Œ v ZiV \ mRM €¿ T¹nb™ l n
¦ § ± — ¸ ZiV l \]B ± ¸ v
© ¸
(2.6)
– FRTL
› §
£
ZiV l \‰F ± £ Z'V l \
¦ § ± — ¸ ZiV l \]B¾© ± Œ v
©
mhM Á¿ ™ l B – ¡
(2.7)
› §
A Define the vector  ZiV[\SBÃÄ ± Z'V[\ , where Å is large enough that HY— ¸ ZiV”"l \8F Å XÆM . A
straightforward computation reveals that
(2.4)
± — ¸ iZ V”l\ B
¥ V
¥
(2.8)
 — ¸ iZ V l \ FI¦ § — ZiV”l\`_UZ'H1— ZiV”l\CF
¥ V
 ¸
Å \  — ¸ ZiV”l\mhMY¡
¸
‹A
¸ ¸ Ž noM
We observe from (2.4)–(2.7) that Ç^È1É A
and ¦ §  — ¸ Z'V l \8moM . Using (2.8), we arrive at
Z'H — ¸ Z'V[\]F Å \  — ¸ ZiV l \msM , which implies
Ç
that  — ¸ ZiV l \moM . Therefore ± — ¸ ZiV l \8B ´ m}M and
we have the desired result.
Another form of the maximum principle for semidiscrete equations is the following comparison lemma.
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ASYMPTOTIC BEHAVIOR FOR NUMERICAL SOLUTIONS
239
£ ¬
Ÿ iZ V[\ tbu v Z[x MYL
ÊË\Lf°
t u l Z'°“ÍΰLf°²\ such that
v \ and Ì b
L EMMA 2.2. Let ± iZ V[\ ,
for V t Z.MYL
ÊË\
¥ — Z'V[\
¥ Ÿ — Z'V[\
±
—
—
I
F
¦
i
Z
[
V
À
\
_
Z
i
Z
[
V
\
L
[
[
V
>
\
O
FI¦ § Ÿ — ZiV[\j_ Ì Z Ÿ — Z'V[\
L[V[\LÏMPnb™n – L
§
¥
¥ V
(2.9)
±
Ì ±
V
Ÿ —fZ'M\ LNM¨n˙n
± —fZ.M\O
– ¡
(2.10)
Ÿ — Z'V[\ MnR™{n
Then we have ± — Z'V[\>O
,
– , V t Z'M(LcÊb\ .
ZiV[\¨B Ÿ{ Z'V[\>F
Proof. Define the vector Â
± ZiV[\ . Let V l be the first V/XÐM such that
ZiV[\XhM
for V t x M(L[V l \ , but  — ¸ ZiV l \]BDM for a certain ™ l¹tκ MYL-¡¢¡-¡L –(» . We observe that
Â
¥ — ZiV”l\
  — ¸ ZiV”l\‰F  — ¸ ZiV”l¯F…½Y\ nhMY¡
 ¸
B ~€$‚
¥ V
¼ l
½
—
¬
l
— ZiV l \`_  —
l
i
Z
V
C
\
F
¸ Œ v ZiV \ mbM €¿ Tnb™+l¹n
¦ §  — ¸ ZiV”l\]B  ¸ v
© Â ¸
– FRTL
› §
ZiV l \‰F  l Z'V l \
¦§  — ¸ ZiV l \]BЩ  v
©
mRM Á¿ ™ l BDMYL
› §
£
ZiV”l\CF  £ ZiV”l\
©
¦ §  — ¸ ZiV l \]B ©  Œ v
mRM €¿ ™ l B – L
› §
‹A
¸ ¸ Ž FU¦ §  — Z'V l \²_ Ì Z Ÿ — ZiV l \
L[V l \8F Ì Z ± — ZiV l \
LfV l \InšM . But this
which implies that Ç^È É A
¸
¸
¸
Ç
inequality contradicts (2.9).
Ÿ 3. Asymptotic behavior. In this section, we show that the solution
of (2.1)–(2.3)
V
goes to zero as approaches infinity and give its asymptotic behavior. To start, let us show
Ÿ{ ZiV[\
that the solution
decays uniformly to zero by the following
Ÿ Z'V[\
T HEOREM 3.1. The solution
of (2.1)–(2.3) goes to zero as V8Ñ«Ê and we have
the following estimates,
MPn
ˆ
Ÿ] ZiV[\ ƒ
ˆ
n
T
Ÿ
v
Y
Œ
K
Zˆ
'Z MJ\ ˆ ƒ
_ËaZ p FbT\”V[\³†‡ „ „
¿iÒÓ
V t x M(Lc_¹Êb\
¡
Proof. Introduce the function Ô ZiV[\ defined as follows,
T
Ô ZiV[\{B Z Ÿ] Z.M\ ƒv ŒYK _baZ FbT\”V[\ †¢‡ „ L
p
„
ˆ
ˆ
k
—
i
Z
[
V
{
\
B
'
Z
[
V
\
Ô
and let Õ
the vector such that Õ
. It is not hard to see that
¥
Õ — Z'V[\
F ¦ § Õ f— Z'V[\j_DZ Õ —kZiV[\f\ K D
B M(LNMnR™]n – FRTLgV t Z.MYL[®G\
L
¥ V
¥ £ Z'V[\
a
Õ
FI¦ § Õ £ 'Z V[\j_DZ Õ £ Z'V[\[\ K “
_ © Z Õ £ ZiV[\[\ d mbM(LWV t Z'M(L[®8\L
¥ V
›
Ÿ f— Z'MJ\
LNM¨n˙n
k
—
'
Z
M
\
m
Õ
– L
Ÿ ZiV[\ ƒ
O
Ê . Setting
where .Z MYL[®G\ is the maximal time interval on which ˆ
ˆ
ZiV[\]B
ZiV[\;F Ÿ{ Z'V[\
and using the mean value theorem, we find that
Â
Õ
¥ —kZiV[\
Â
F‘¦ §  —fZ'V[\j_ p Zi֍—fZiV[\[\+KŒ v  —kZiV[\]BUMYLWM¨nb™{n – FRTLWV t 'Z MYLf®8\
L
¥ V
¥ £ ZiV[\
a
Â
F‘¦§  £ Z'V[\j_UZ p Z'Ö £ Z'V[\[\ KŒ v _ © ZiÖ £ ZiV[\f\[d Œ v \  £ ZiV[\mhMYLgV t 'Z MYLf®8\
L
¥ V
›
 — Z.M\mhMYLWM¨n˙n – L
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240
N. DIABATE AND T. K. BONI
Ÿ —kZiV[\
and Õ —fZ'V[\ . Apply Lemma 2.1, to obtain
we have
T
Ÿ{ Z'®8\ ƒ n
ODÊDL
ˆ
ˆ
Ÿ
Zˆ
Z.M\ ˆ ƒv Œ(K _baZ p FRT"\^®8\ †‡ „ „
which leads to a contradiction. Hence ®DBoÊ and we have the desired result.
where ֍— is an intermediate value between
M¨n Ÿ Z'V[\>n Õ iZ V[\ for V t Z.MYLf®8\ . If ®}O|Ê
The statement of the main result of this section is the following
Ÿ T HEOREM 3.2. Let
be the solution of (2.1)–(2.3). Then we have
~$€
Ÿ] ZiV[\ ƒ B ul L
A'‚wƒ V ’^‡ „ „ ˆ
ˆ
where ul B“Z Š[‹ d v Œ v^Ž \ ’^‡ „ „ .
The proof of Theorem 3.2 is based on the following lemmas. The function
×]Z'Q\{BØF Å Z u l_…Q \j_baZ u l_³Q\ d L
where u l3BÙZ Š[‹ d v Œ v[Ž \ ’^‡ „ „ and Å B d Œ v v , is crucial in the proofs of the lemmas below. The
Ÿ ZiV[\
of (2.1)–(2.3) is bounded from above by a
following result show that the solution
function which decays to zero.
Ÿ{ L EMMA 3.3. Let
be the solution of (2.1)–(2.3). For any Ú XØM , there exist positive
®
times and Û such that
Ÿ — ZiVj_
Û \²nsZ u{l _ Ú \Z'Vj_…®8\ Œ Ä _ËÜ — Z'Vj_…®8\ Œ Ä Œ v LNM¨nb™]n – L
Š
where ÜC—ÀBªF § Z u l>_ Ú \ d ™ § › § .
Proof. Introduce the vector Õ
defined as follows,
Ä
Ä
u
k
—
i
Z
[
V
]
\
“
B
Z
l
_
Õ
Ú \”V Œ _³ÜC—.V Œ Œ v LNMPnb™{n – ¡
A straightforward computation reveals that
¥ —
Õ
K
u l _ Ú \”V Œ Ä Œ v FRZ Å _DT"\^V Œ Ä Œ § Ü —
¥ V F…¦ § Õ — _ËH Õ — B}F Å Z _|HJV Œ Ä K Z[Z ul _ Ú \j_ËÜ — V Œ v \ K FIV Œ Ä Œ v ¦ § Ü — L
¥ £
a
£ _ËH
£
Õ
K£ _“©
d£ B}F Z u l>_ \”VcŒ Ä Œ v _³HJVcŒ Ä K(ZfZ u l>_ `
¥ V F‘¦ § Õ
Õ
Å
Ú
Ú \ _³Ü VcŒ v \%K
› Õ
FoZ Å _DT"\”V Œ Ä Œ §"Ü £ FÎV Œ Ä Œ v ¦§-Ü £
a
_Ý© V Œ Ä Œ v Z[Z u{l _ Ú \j_ËÜ £ V Œ v \ d L
›
because Å q¹B Å _|T . From the mean value theorem, we get
Z u l_ Ú _ËÜ £ V Œ v \fd¯BªZ u l²_ Ú \[d]_³Þ £ ZiV[\^V Œ v L
£ ZiV[\
is a bounded function. We deduce that
¥ —
Õ
K
_ T\”VcŒ v ÜC—
¥ V F‘¦ § Õ —2_ËH Õ — B|VcŒ Ä Œ v Z'×]Z Ú \CFhZ Å |
¬ ¬
_DHJVcŒ Ä K Ä v Z u l_ Ú ³
_ VcΠv C
Ü —%\+Ke\L
¥ £
a
£ _ËH
Õ
K£ _ ©
d£ B|V Œ Ä Œ v Z[F Z'×]Z \
¥ V F‘¦§ Õ
Õ
Å
Ú
› Õ
¬ ¬
_DHJV Œ Ä K Ä v Z ul _ Ú ³
_ V Œ v Ü £ \K _
a
_ © Þ £ V Œ v \¡
›
where Þ
©
a
›
Z ul _ Ú \[d
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ASYMPTOTIC BEHAVIOR FOR NUMERICAL SOLUTIONS
241
We observe that ×]Z.M\{BDM and × z Z'M\]BØT . This implies that for Ú XhM , ×]Z Ú \XRM . We also see
K ORM . We deduce that there exists a positive time ® such that
that F Åep _ Å _DT8B dd ŒY
Πv
¥
Õ — F ¦
¥ V I
¥ £
Õ
¥ V FI¦
§ Õ — _DH Õ — K XhMYL
a
§ Õ £ _DH Õ K£ _ß© Õ d£ XhMYL
›
® Œ Ä ul
¡
Õ —fZ'®8\X
©
Ÿ Z'V[\
X ® such that
Since from ‡Theorem
3.1
goes to zero as V tends to infinity, there exists Û b
"
à
á
Ÿ — Z \:O ¤
ZiV[\
ZiV[\B Ÿ{ Z'V`_
¸ O
—
'
Z
8
®
\
. Introduce the vector Â
such that Â
Û
Õ
Û FI®8\ . A
§
routine computation yields
¥ —
Â
K
¥ V F‘¦ §  — _|H  — XRM(LWM¨nb™]n – FRTLWVmb®:L
¥ £
a
£
Â
£K
£
¥ V F‘¦ §  _|H  _ß© ›  d ZiV[\>XRMYLWVmË®¯L
Ÿ —fZ \>O
 —[Z'®8\]B
Û
Õ —kZi®8\¡
ZiV[\²n
We deduce from Lemma 2.2 that Â
Ÿ —fZiVj_
Z'V[\
. That is,
¿iÒÓ VmR®¯L
Õ —kZiV[\
Õ
Û FI®8\n
which leads us to the result.
Ÿ{ L EMMA 3.4. Let
be the solution of (2.1)–(2.3). For any Ú XhM , there exists a positive
time Û such that
Ÿ — ZiV`_|T\msZ ul F \¢ZiVj_
Ä
Ä
Ú
Û \ Œ _³Ü — ZiVj_ Û \ Œ Œ v LNMPnR™{n – ¡
Proof. Introduce the vector ±
such that
Ä
Ä
± — ZiV[\BªZ ul F Ú \”V Œ _ËÜ — V Œ Œ v ¡
A direct calculation yields
¥ —
±
K
¥ V F‘¦§ ± — _ËH ± — BØF Å
_UHZfZ
BDV Œ Ä
_UHZ u
¥
±
¥ V
£
Z u{l F Ú \”V Œ Ä Œ v FhZ Å _|T\”V Œ Ä Œ §
ul F Ú \”V Œ Ä _³Ü — V Œ Ä Œ v \ K
Œ v Z^F Å Z ul F Ú \‰FRZ Å _DT"\”V Œ v
l¯F Ú _³ÜC—.VcŒ v \%K\
L
a
u l F Ú \”V Œ Ä Œ v FhZ Å D
F‘¦ § ± £ _ËH ± K£ _ß© ± £ d BªF Å Z _ T"\^V Œ Ä Œ § Ü £
›
_DHJV Œ Ä K Z ul F Ú _³Ü £ V Œ v \ K I
F V Œ Ä Œ v ¦ § Ü £
a
_
V Œ Ä Œ v Z u{l F Ú _³Ü £ V Œ v \ d L
© ›
£
because q Å B Å _ÝT . From the mean value theorem, we have Z u>l F Ú _ØÜ V Œ v \ d B
£
£
Z ul F Ú \ K _RÞ iZ V[\^V Œ v , where Þ ZiV[\ is a bounded function. Using this equality, we deduce
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242
that
N. DIABATE AND T. K. BONI
¥
± —
K
_ Þ — V Œ v \
L
¥ V FI¦ § ± — _³H ± — BhV Œ Ä Œ v Z'×]Z^F Ú \‰FRZ Å _DT"\[Ü — V Œ v
¥ £
a
£
±
£
K£
\ FRZ Å D
_ T"\^V Πv
¥ V F‘¦ § ± _ËH ± _ß© › ± d BhV Œ Ä Œ v 'Z ×]Z^F Ú ‰
a
¬ ¬
_8⠗ V Œ v _ © V Œ d Ä Ä v Z ul F Ú ³
_ V Πv \ d \
¡
›
Obviously, F¯q Å _ Å _}TSOM . Also, since ×]Z'MJ\rBÝM and × z Z'MJ\rBÙT , it is easy to see that
×]Z^F Ú \OhM . Hence there exists a positive time ® such that
¥ —
±
K
¥ V F‘¦ § ± — _ËH ± — ORMYLNMPnR™{n – FbTLgV t x ®¯Lc_¹Êb\
L
¥ £
a
£
±
K£
£
¥ V F‘¦ § ± _ËH ± _ß© › ± d ObM(LWV t x ®:Lc_¹Êb\
¡
Since ±
± Z Û \²O
ãeä Z'®¯L¢T\
iZ V[\
to zero as V approaches infinity, there exists Û X
such that
Ÿ Z[goes
T"\ . Setting å ZiV[\]B ± ZiVj_ Û FRT"\ , we observe that
¥ —
å
K
¥ V FI¦ § å —‰_DH å — XhMYLWM¨nR™]n – FbTLWVmb®¯L
¥ £
£ _DH £ K _ß© a d£ XRMYLWVmË®¯L
å
¥ V FI¦ § å
å
› å
Ÿ —[Z[T"\LNM¨n˙{n
k
—
^
Z
T
]
\
B
f
—
Z
>
\
O
å
± Û
– ¡
We deduce from Lemma 2.2 that
Ÿ — ZiV[\m
± — ZiVj_ Û FRT"\
for V>mUTL
which leads us to the result.
Now, we are in a position to prove the main result.
Proof of Theorem 3.2. From Lemma 3.3 and Lemma 3.4, we deduce
Ÿ
Ÿ —[Z'V[\
$ €µY¿ Z —fZ'V[\ \>n ~$€
Z u{l F Ú \n A'~€‚w
Z
\n}Z ul _ Ú \
L
ƒ
'
A
w
‚
ƒ
f
æ
Y
ç
è
VÄ
VÄ
for any Ú XRM and the proof is complete.
4. Convergence. In this section, we will show that for each fixed time interval x MYLf®²y
Ÿ ZiV[\
where @ is defined, the solution
of (2.1)–(2.3) approximates @ , when the mesh param›
eter goes to zero.
·
T HEOREM 4.1. Assume that (1.1)–(1.3) has a solution @ tSuGé v Zfx M(L¢T¢y`ÍIx MYLf®²yê\ and the
initial condition at (2.3) satisfies
Ÿ l FÎ@ Z.M\ ƒBUë0Z^T\gH0ìí›!њM(L
(4.1)
ˆ
ˆ
£
£ for › sufficiently small, the problem (2.1)–
where @ ZiV[\BšZ'@‰Z'Q l L[V[\
L-¡€¡$Ÿ ¡$L[@CZiQ fL V[\[\^¤ . Then,
t9u v Zfx M(L[®²y+Lf° ¬ v \
(2.3) has a unique solution
such that
l
ãä
Ÿ] Z'V[\CFI@ ZiV[\ ƒ BUî¨Z Ÿ FI@ Z.M\ ƒ _b›2§-\gH0ìí›!ѐMY¡
l-¶2A+¶ ˆ
ˆ
ˆ
ˆ
¤
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ASYMPTOTIC BEHAVIOR FOR NUMERICAL SOLUTIONS
·
Proof. Since @ tué v , there exist two positive constants ï
@ E-E"E ƒ
© ˆ ñ ˆ
ï
ˆ @ E-E"E-E ˆ ƒ
T
©
ï
243
and ð be such that
v
ˆ @ ˆ ƒ6n ï LNH p Z ï _DT"\+KŒ n ð L
q0Z _DT"\ d Œ v n ð ¡
(4.2)
© ï
Ÿ tSu v Z[x MYL[® \
Lk° £ ¬ v \
The problem (2.1)–(2.3) has for each › , a unique solution
. Let VZ.›\
d
V
h
X
M
the greatest value of
such that
Ÿ Z'V[\CFI@ ZiV[\ ƒÐOUT ¿iÒÓ V t Z'M(L[VZ.›\[\¡
(4.3)
ˆ
ˆ
¨$µ º VZ.›\
L[®
» .
The relation (4.1) implies that VZ.›\8XUM for › sufficiently small. Let V Z.›\B
By the triangle inequality, we obtain
¿iÒÓ V t Z'MYLfV Z%›2\f\
L
Ÿ ZiV[\ ƒ¾n
Ÿ ZiV[\CFI@ ZiV[\ ƒ
ˆ
ˆ
ˆ @‰Z'QjL[V[\ ˆ ƒ}_ ˆ
ˆ
n
L
©
n
L
©
which implies that
(4.4)
ˆ
Ÿ] ZiV[\Fb@
Let à ZiV[\rB
t
V
Z
'
M(L[V Z%›2\f\ ,
have for
¥ à — ZiV[\
FI¦ § à — ZiV[\]B
¥ V
¥ Ã £ ZiV[\
FI¦ § Ã £ ZiV[\]B
¥ V
Ÿ] Z'V[\ ƒ
ˆ
nsT_ ï
¿iÒÓ
V t Z'M(L[V Z.›\[\¡
ZiQÀL[V[\
be the error of discretization. Using Taylor’s expansion, we
› §
@ -E E"E-E .Z Qò — [L V[\‰F‘H p ⠗KŒ v à — Z'V[\
L
T
©
› §
› §
© qÖ £ d Œ v à £ _ © ñ @ E-"E E Z.Qò £ [L V[\j_
@ E"E"E-E Z.Qò £ L[V[\‰F‘H p â £KŒ v à £ ZiV[\L
›
T
©
Ÿ —[Z'V[\
Lf@‰Z'Q2—kL[V[\
£ t Z Ÿ £ ZiV[\
Lf@‰Z'Q £ L[V[\
t
Ö
¢
â
—
Z
where
and
. Using (4.2) and (4.4), we arrive at
¥ à — ZiV[\
F‘¦§¢Ã—[Z'V[\>n ð8ó Ã"—[ZiV[\ ó _ ï ›2§LNM¨n˙n – FRTL
¥ V
¥ Ã £ ZiV[\
£
£
Z Ã
Z'V[\CF Ã £ ZiV[\f\
£
ð8ó Ã ZiV[\ ó _
Πv
F
©
©
n
¥ V
ð8ó à ZiV[\ ó _ ï › § ¡
› §
›
‹i‹$õ ¬ v[Ž Ai¬ á E-ö Ž Z Ÿ l F³@ Z'MJ\ ƒ _D÷w› § \
Consider the function ô Z'QjL[V[\BÃ
, where ø , u , ÷
ˆ
ˆ
are constants which will be determined later. We get
ô A ZiQÀL[V[\;F ô E-E iZ QjLfV[\{BªZ ø _|T²F © u
ô E 'Z M(L[V[\]B|M(L ô E Z[TLfV[\]B ©
á E-ö Z Ÿ l FI@
ô Z'QjLkM\]B|Ã
ˆ
FÎù u § Q § \ ô Z'QjL[V[\L
u ô Z[TL[V[\L
Z.M\ ƒ
ˆ
_b÷w›\
¡
By a semidiscretization of the above problem, we may choose ø , u , ÷ large enough that
¥ ZiQ — L[V[\
ô
Xh¦ § ô ZiQ — LfV[\`_ ðGó ô ZiQ — LfV[\ ó _ ï › § LNMPnR™{n – FRTL
¥ V
¥ ZiQ £ L[V[\
ô
Xh¦§ ô ZiQ £ LfV[\j_ ð ó ô ZiQ £ LfV[\ ó _ ðGó ô ZiQ £ LfV[\ ó _ ï 2
› §eL
¥ V
›
ô ZiQ—[LkMXh×[Z.M\
LNM¨nb™{n – ¡
It follows from Lemma 2.2 that
ô ZiQ — LfV[\XRà — ZiV[\
¿iÒÓ
V t Z'M(L[V Z.›\[\
LúM¨n˙{n – ¡
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244
N. DIABATE AND T. K. BONI
In the same way, we also prove that
¿iÒÓ
ô ZiQ — LfV[\XoF¯Ã — Z'V[\
which implies that
Ÿ{ Z'V[\CFÎ@ Z'V[\ ƒ
ˆ
ˆ
V t Z'MYLfV Z%›2\f\
LNM¨nR™]n
Ai¬ á
l
Ž Z Ÿ FI@ Z'MJ\ ƒ
nRà ‹$õ
ˆ
ˆ
– L
_b÷w› § \
LWV t Z'M(L[V Z.›\[\
¡
Let us show that V .Z ›\{Bh® . Suppose that ®}XËVZ%›2\ . From (4.3), we obtain
¬
l
T:B ˆ Ÿ ZiVZ%›2\f\;FI@ ZiVZ%›2\f\ ˆ ƒÐnhà ‹$õ ¤ á Ž Z ˆ Ÿ FÎ@ Z'M\ ˆ ƒ}_Ë÷r› § \
¡
Since the right hand side of the above inequality goes to zero as › goes to zero, we deduce
that T¹nhM , which is impossible. Consequently V Z%›2\Bh® , and we obtain the desired result.
5. Numerical results. In this section, we give some numerical
results. First, we apŸ ‹€û Ž BÝZ Ÿ l û L Ÿ û L-¡¢¡-¡¢L Ÿ £ û \^¤
of
proximate the solution @CZiQÀL[V[\ of (1.1)–(1.3) by the solution
v
the following explicit scheme,
Ÿ — ‹€û ¬ v^Ž F Ÿ — ‹€û Ž
‹€û
‹€û
‹€û ¬ v[Ž
B|¦ § Ÿ — Ž F‘HZ Ÿ — Ž \ KŒ v Ÿ —
LNM¨n˙n – FRTL
ü V
Ÿ £ ‹€û ¬ v^Ž F Ÿ £ ‹€û Ž
a
‹€û
‹€û
‹€û ¬ v[Ž
‹€û
‹€û ¬ v[Ž
B|¦ § Ÿ £ Ž F‘HZ Ÿ £ Ž \+KŒ v Ÿ £
F}© Z Ÿ £ Ž \ d Œ v Ÿ £
L
ü V
›
Ÿ — ‹ l Ž BDý — XhMYLWM¨nR™]n
– L
ö
ü
where þ mßM , VPn § . Let us remark
that the restriction on the time step guarantees the
Ÿ ‹€û Ž
positivity of the discrete solution
.
Ÿ ‹€û Ž
Now, approximate the solution @‰Z'QjL[V[\ of (1.1)–(1.3) by the solution
of the following implicit scheme,
Ÿ — ‹$û ¬ v[Ž F Ÿ — ‹€û Ž
‹$û ¬ v[Ž
‹$û
‹€û ¬ v^Ž
BD¦ § Ÿ —
FhZ Ÿ — Ž \ KŒ v Ÿ —
LÿT¹nR™]n – FbTL
ü V
Ÿ £ ‹$û ¬ v[Ž F Ÿ £ ‹€û Ž
a
‹$û ¬ v[Ž
‹€û
‹€û ¬ v^Ž
‹$û
‹€û ¬ v^Ž
BD¦ § Ÿ £
F‘H Z Ÿ £ Ž \ KŒ v Ÿ £
F}© Z Ÿ £ Ž \ KŒ v Ÿ £
L
ü V
›
Ÿ — ‹ l Ž BUý2—]XRM(LNMnR™]n
– L
where þ mRM .
The above equations may be rewritten in the following form,
ü V ‹€û ¬
ü V
‹$û
‹€û ¬ v[Ž
Ÿ — ‹€û Ž BªF
Ÿ —
v^Ž _UZ^T_
_³H ü VZ Ÿ — Ž \ KŒ v \ Ÿ —
F
© › §
› §
Πv
ü
ü
V
‹€û
‹€û ¬ v[Ž
Ÿ l ‹€û Ž BªF © V Ÿ ‹€û ¬ v[Ž _UZ^T_
_ËH ü VZ Ÿ l Ž \ KŒ v \ Ÿ l
L
v
© › §
› §
ü
ü
V
a
‹€û
Ÿ £ ‹€û Ž BªF!© V Ÿ £ ‹€û ¬ v[Ž _UZ^T_
_ËH ü VZ Ÿ £ Ž \ KŒ v _ß© ü V ó Ÿ
v
©
› §
›
›
Œ
§
which gives the following linear system,
‹€û Ž Ÿ €‹ û ¬ v^Ž B
where
Ÿ €‹ û Ž L
‹€û Ž
is a tridiagonal matrix defined as follows,
ü V ‹€û ¬
Ÿ $— ¬ ^v Ž L
v
› §
¬
£ ‹€û Ž d Œ v \ Ÿ £ ‹€û [v Ž L
ó
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245
ASYMPTOTIC BEHAVIOR FOR NUMERICAL SOLUTIONS
¥ l
A
Œ ö
M
‹€û Ž B
..
.
M
M
Œ §ö
¥
v A
Œ ö
..
.
M
M
M
A
M
M
Œ ö
¥
M
A
§
..
.
M
A
Œ ö
..
.A
Œ ö
M
M
M
M
M
M
M
..
.
¥ £
Œ A§
Œ ö
M
..
.A
Œ ö
¥ £
ΠvA
Œ § ö
M
..
.
M
A
Œ ö
¥ £
L
A
ü
¥
Ÿ €‹ û Ž
with — BØT_ © ö _ËH VZ — \ K Œ v for M¨nR™]n – FbT and
ü V
a
‹€û
‹€û
¥ £ B}T_
_ H ü V ó Ÿ £ Ž ó KŒ v _ß© ü VZ Ÿ £ Ž \ d Œ v ¡
³
© › §
›
‹€û Ž
Let us remark that the matrix
satisfies the following properties
‹€û Ž
‹€û Ž
‹€û
€‹ û Ž
OhM ¿iÒÓ ™ B L ó —€— Ž ó X —€— XbM(L
— ó — ó ¡
— Ÿ û
Ÿ ‹€û Ž mbM
These properties imply that
exists for l any þ and
; see, for instance,
ö [3].
ü
Ò
Ÿ
ñ
We let p B , q¹B © , HPBªT , a²B}T , — BUMY¡ {_SM(¡ æ Zj›Y™^\ and V;B § . In Tables 5.1
and 5.2, in the rows, we give the first þ for which
ü Ÿ ‹$û Ž FRT ƒÝO
Ú BØT-M0Œ § L
ˆ þ V
ˆ
ü
the corresponding time V û B þ V , the CPU time, and the order Z.ì\ of method computed from
~€
Ò ZfZi® Fή \kž0Z'® FI® \[\
é
§
§
ì8B
¡
~€Ò Z \
©
TABLE 5.1
Numerical time, number of iterations, CPU time (seconds), and order of the approximations obtained with the
implicit Euler method
16
32
64
128
256
"!
#
$&%'¨)(+*-,
.
301.1953
303.1289
303.1140
303.1177
303.1186
154211
620807
2483229
9932910
39731654
43
319
1017
9237
13515
2.16
2.02
2.04
TABLE 5.2
Numerical time, number of iterations, CPU time (seconds), and order of the approximations obtained with the
explicit Euler method
16
32
64
128
256
!
#
$&%'¨)(+*-,
.
301.1953
303.1289
303.1140
303.1177
303.1186
154211
620807
2483229
9932910
39731654
43
319
1017
9237
13515
2.16
2.02
2.04
ETNA
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246
N. DIABATE AND T. K. BONI
REFERENCES
[1] L. A BIA , J. C. L ÓPEZ -M ARCOS , AND J. M ART ÍNEZ , On the blow-up time convergence of semidiscretizations of reaction-diffusion equations, Appl. Numer. Math., 26 (1998), pp. 399–414.
[2] T. K. B ONI , On the asymptotic behavior of solutions for some semilinear parabolic and elliptic equation of
second order with nonlinear boundary conditions, Nonlinear Anal., 45 (2001), pp. 895–908.
, Extinction for discretizations of some semilinear parabolic equations, C. R. Acad. Sci. Paris Sér. I
[3]
Math., 333 (2001), pp. 795–800.
[4] V. A. K ONDRATIEV AND L. V ÉRON , Asymptotic behaviour of solutions of some nonlinear parabolic or
elliptic equation, Asymptot. Anal., 14 (1997), pp. 117–156.
[5] R. E. M ICKENS , Relation between the time and space step-sizes in nonstandard finite difference schemes for
the Fisher equation, Numer. Methods Partial Differential Equations, 13 (1997), pp. 51–55.