View of On the numerical solution of a nonlinear parabolic equation

MATIIÞMATICA

_

REVUE D'ANAITYSE NUMÉRIQUE ET DE THÉORIE DÞ IT'APPROXIMATION

L'ANALYSE

^NUMÉRIQUE

ET LA THÉORIE DE

L'APPROXIMATION Tomo B, No

1'

1979, PP.

B9-97

ON THE NUMBRICAI, SOI,UTION OF A ^NONLINEAR

PARABOI,IC BQUATION

'by

ERVIN ^SCIIECHTER

(Cluj-NaPoca)

e

conditions under

which the

expli-

a

solution converging

to

the

very restrictive and

^exclud'e

f viéw. It is the aim of

the under much

larger

and natu-

l.Formulationoftheproblems.Conçiderthefollowingproblem:

(1.1) *: ^{orlø) oo ?} : o X

_10,

1[

(1.2) u(x,'0):uo(x) ^ßeÇ2

(1.3) ^{u(x, t)} : ut(x, t) oo

^S

: o x [0' T]' As in _[5]

we ^suppose

the

following assumptions

to

hold ^:

(i)

øo

e C(o)

^uv

e C(S),

^uo,

^{ut 2}

⁰

(A) (iÐ ltq e

Cz(Ra),

9@)

^{anð' 9'(u)}

> ⁰ for

ø

>

⁰

ç(0)

:

_9'(0)

: 0, 9"(u) Z0 ^{fot u à}

^0'

we also

sllppose

that o (Rr,(for sirnplicity)

^and

that'this

^{domain is}

bounded, regular and convex.

s0

(1.5) (1.6) (r.7)

We

called

a function u e L-(Q) a

solution

of (1.1)-(l.B) if

(i) # ⁼ r,e) i: ^r,

²

(ii) conditions (1.2)-(1.3)

are

fulfilled (in the

generalized sense)

(ili)uZ0

(a.e.)

onQ

(iv) For

any

f -

^{C'(Q) such}

^that

,/1",

:

0 (1.4)

3 ON THE NUMERICÀL SOLUTION 91

IIere

(l;(h)

:

^u(h)

-

^y(k

-

¹⁾ _'

uo¡:

^üolQn,

^üzi l, X

_{Þoc'

ho:0' ^1' "

^'

' ^K} -'

R'

ur(x,

t)

Í: ^{ut(x*, t),} x e|'r,,

where

x* e

^ðÇì

^{is the}

^neafest

^{point to ø (or}

^one

^{of them,} if there

are

mo

^ls).

M a

^posítive coltstalrt such

tirat

üo' nd

exact solutions we

shall

make

use

- of the

^following extensions:

a)

Constant exÌension.

This

attaches

to

each

function [/

defined on

the grid points of _Q¡ (or Or) a step fu'ction Û

^d.efined

^{on the}

^domain

Q.CQ ^{(or Qr) by:}

U(x, t) : U¡¡(h) ^(x, t) e

^,,i,¡j

x fhr'

^(Þ

f ^1)r['

b) Multilinear

extension.

This

^assigns

to the grid ^function IJ a

con-

tinuoús function U'

defined bY:

U':QrnR

U'(x, t) :

U¡¡(h)

¡

(Ut¡(h))-,(x,

- ^ih) +

(Ui¡(h))',(x2

- ^{iD +} a

(Ur;(å)),(t

-

^kr)

¡

(U1(k)),,",(xt

-

^ih')(x,

- ^{tD +} . + (U4@)),,t(4:- ih)(t - k) + (Ui¡(h)),,t(xz- il')(t -

^k'),

(x, t) e

^lo¡¡

X \tu,

^(Þ

f ^1)t['

For a

^detailed.

^{study of}

these extensions we refer

to _[3]'

2.Basiclern¡nas.In[5]weprovedthefollowingtwolernmas:- I,em ma

^{2.1. Suþþòse}

^that

assumþtion

(A)

hold's ønd' tkat

^:4 ^{*v'(M) =t.}

TkentkesolutionUof(1.5)-(1.7)sarisfi'esth'e'inequøl'ity:

0su<M.

I,

e m m

a 2'2,

Suþþose

tkat

the hyþotkeses

of

Lemmø

2.|

^are fulfill'ed' and assume tkøt:

(i) u¡ e

^{C\(Q) ønd' A'u,}

^Z

⁰

^on

^dl-;

iii¡ *t ^is

nònd'ecreasing-

in t on

10, T I

ERVIN SCHECHTER

2

ôç(u) ôx¡

_af ôt¿

2

af

D

At

u d,xdt

f

.f

(x,

0) uo@) d.x

:0

a Õ

HereSr:SU{*,T)lx=Q}.

Remørh,.

condition

(1.2) makes sense. rudeed,

it wilr

be shown

that 9(u)

^belongs

to Z(0, T;

^111(C))). So

by

(1.1)

it follows that L _{e L(0,}

_T;

l^-'(9))._ThÞ

_1.2,

_ch. r). we

^implies also

note that u ^that u e C([0,

has *é11 defined

Tl; Ht(a)),

tràces (see

e.s. oä 5 l4]. ivhich

^f,emmade-

pend

continuously

on I

^(e.g.

_l3l p.

70).

pro lu.,'l'='Bi;i; l;ffiä,'äî;f J8'i#; î'ö:'?i

ord

explici-t_numerical sct

"-"

usdoôiat"a

to (1.f

^(1.8)

we from

_[5] :

R¡: _{(x, t) e Rslxr:

h¿

h, t :

ho.ç,

k;:0, _{+.1, +2,}

^. .

., i:1, 2;

ko:0, 1, "').

where

h, , > ⁰

^are th.e mesh-sizes,

.oij

: _{(xu

xr)l

ih<x, < ⁽ⁱ + ^{t)h, jh} <

^xz

< (i + ^t)k}

O, :

_LJ^.¿

j, lt -

^{ôO¡, Q,,}

:

Qn

X

_10, ^7-].

We shall

use

the *1"

""rgra" -a, ^the

mesh-points

of R,

belonging to these. domains,

e.g. Q¡: Q¡ î\Rn, ^{etc. F'or brevity} *" shull jso"set

!,¡(h) or eve'

^{U(,ë) instead of.}

^{U(ih, jh,, tl'').}

^Now

^the

difference problem

is the following: '

^'

Ui(h)

:

Are(U(Þ

- ^{1)) or}

^Qn

u(0) :

u"h

U(k)lrr: üz(*, hr), x el¡, h:0, 1, ..., K:

^T

I

92 ^{ERVIN SCHECHTER}

Then

(j) u,(h) > 0, h : l, 2, ..., K on

^O

(ij) "to'Eu#l { ^Mm($.

Qh

IIere

Õr:0,ì\{(r, ^K.)l.x =

^Õr}.

Remørh.

Condition

L,uo

2 0 can be

^replaced

^by have already

mentioned,

conditions (i), (ä)

are too therefore prove:

Lemma2.3.

Suþþose tkøt

(i)

Asswmþtion (.L) kold.s a.nd.

u, e

Cz(Q);

(ii)

À

( l;

¡i11¡.-lf-

\/ðt

exists and is bounded

on S;

(iv)

g(co)

:

oo.

Tken tkere exist constants ko, C

> ⁰

^{suck tkøt}

(2.r) ,k'ÐlUlk)l {

^C,

for k x

^tt,o.

Q¡

Proof

: Let e>0beaconstant ^anð.V¡: C)*R a functionsuchthat

(2.2)

^LVo

^2lAe(øo)l f e on

^O,

VoÈ

Z q@o)lr,'Vo2 0, Vo =

^e¡@).

We define, aoby vo

:

g-r(Vo)and u, : S

-*'R

as follows : u1

è

^{0, u1 G C(S),} 7arlôt exists

on S

and

*@, ^{t) >}

^c

I

^e,

t _{e lo,} 1[,

^ar(x,

o):

^r/olr

ôt'

where

c àmaxpur/ðtl.

With the

s

aíd of to, at as data

functions we construct

by

^means

of the

scheme

(1.5)-(1.7) the

discrete solution

'V:Q,+R'

Thus taking into

account

the

above

Renark

we-

get by

l.emma

2.2

tine following inequalities

and

v¡ Z ot

Qr

'ckz

Ðvlu¡ ^{> M'm(a)}

Q,r

b ^{0N lHE NUMERICAL sÔLUrloN}

For

convenienceweset U;(Þ)

:

Ú(h)

andlet ^us

show

that

D

ltTdi(l)1,<

D

^7,,(1)

T^

^o[

rndeed

by (2.2) lor h1ho,

^Lu1øo)

>

lA¡ç(øo)l' Hence according

to

^(1'5)

we

get

(2'3).

-SupPose now

that

)ì

^lüd,(Þ

- ^{1)l < DZü(h} -

^r)

th

^oå

93 4

(2.3)

aq(øo) >

restrictive,

0. As

we we shdll

(2.4)

and prove

(2.5)

flr7r,(Þ)l .õl',h <DV;¡(h)' From

^(1.5)

^we

^{deduce tlnat}

(k Z

²⁾

(2.6)

, , U¡(k):U¡(P-,l) +Ar(ç(Þ-

¹⁾⁾

- ^e!(h'2))'

consequentlY

(2,7) ^Úi¡(h):

⁽¹

_{-4 #r,oþ-}

^r))Úq(h

⁺

¹⁾

⁺

+ h,lói+r,,(Ë -

^l)

^{t;¡r'¡(Þ- l) + 4;-t'¡(h} - ^l)Ú;-ui(å - ^{t) +} I ^4i,¡*r(h -

1)Ú¡,r+r(h

- ^{r) *}

^,r,,-.rru

- r)tt,¡-r(A -

^1)l

and similarly for V.

^Here _-Q.'

^means an intermed'iary value

^between

[1U(å-f ijj'""a

e'(U(h

- ^{2))' Foe}

^example:

!i¡(k

- ^t) :

_e'(Ut¡(h

-

²⁾

^{+ s(Uü(h-}

¹⁾

-

^Uû(h

-

^2)))'

0 e

_10, ^1['

From

(2.7)

we get

^(Þ

à

^1),

D|tr,,(¿)l sÐlÚ,i(Þ ^1)l +;Ðe'(tlt¡(k - l))lÚ¡¡(Þ-

^1)l

q,'

Dv,¡(H

^S

Ð

^7,¡(e

- ^{l) +} ]fiv'Ú',(h -

^r))Vq(Þ

^-

^1)'

-dùr' 4, ^h'

^rn

Since uol¡

è

^uolr _^nð,aff

I

^Lu'

^{on S;}

^{ho can be}

^taken

^{so small}

^that

ur(r,

t* _{") S}

^ut(x'

^{t) if t S .o} : ffi'

j: !i

!r ,¡

."r' i

ì:, :r'

g4

in this

case

for ^k <

^ho.

ÞRVIN SCHÉCHTER ON THE NUMEiìICAL SÓLÚTÍON 96

6

I

A'i(þ'- l) _<

7rr(Ë

- ^{1) on} l,

which

irnplies

e'(l¡¡(h -

¹⁾⁾

^<

^e'(Tn¡(h

- ^{1)) on lr.}

Here

.t¡¡.(h.- l)

=.U,¡(h -2) +

^O(Uti(Þ

- ¹⁾ -

^U¿¡(k

-2)), ⁰

^{,.=- ,10, 1[}

and

similarly lor

V ¿i.

On

the other hand

since ôarf 0t

7 ^{c on S} ít

follows

that Itü(/è)l S

^Z;¡(p)

^on t¡, h:O, t, ..., K.

Ifence

lv'(I,i(t, rn - 1))ltn¡(h- t)l <Ð -q' ' ç'(7¿¡@_ r))Vtj(h-L)

^'r\

'so _{th_at}

(2.4) implies (2.5). Finally, if we

observe

that on the

discrete boundary

oï

_Ç

Dlu;(A)t S Dvlnl.

our lemma follows

at

once from Iremma 2.2.

Corollary.

Under

the

hyþotheses

of

the aboae Lemrnø:

(2.8)

"k,Ðl,p(u(h)))-tl <

^c,

ah

C being ø constønt indeþendent of

k

(andr).

This is readily

seen from

þQ

@)')

:

_ø' (h)lu;(À)

I S

e'

(M)lui(h)'l

and Iremrna 2.2.

rn

order

to

estimate

the

differences

in the

space variables we give:

Lemma2.4. Let !_,QonV^be ^{the solution} of (l.S)-(1.7)

and suþþose

lhat

cond.itions of Lemma

2.8

ørc futfilted..

Tken tkere exist

constønts

c,

ho,'ind,eþend.ent

of k (andt)

suck tkat

Here

Ql.

is

a þotygona.l d'omain

uitk

^si'des þørallel to ^Oxr,

Oxr;

conuex

in

both d,irections ønd' sati,sfying lhe condilions ^:

(u) Õi c ^Q*,

^ç¿*

c ^a

(l)

^{The sid,es}

^of

^{âQl, contain nod'es}

^of

^R,,'

The proof

^{becomes identical}

to that o! _t5J

(Theorem 3.1),

if

^instead

of

l,emmã

2.2

we make use

of

Lemma ^2.3.

I,em rr.a2.5. ([1],

Theorem2.22)

Iel 1( _þ <?

^ønd'

let-Kç-!''(9)'

Suþlose there ^exists

à'ttqutnrtlQ¡\ oÍ

subd.oma.ins

of

_Q ^{hauing the} follouiøg þroþerties ^:

(l) For

^each

j, _Q¡CQ¡+ti

(ä) For

^each

j the

^set.

of

restrictions _{to Q¡}

of

the functions

'in K

^is

þrecornþact

in

Lo(Q¡) ;

(äi) For

eaery e

2 ^0,

there exists

j

suck thøt

\ t"tùlþ

^d,x

<

^e

for

^eaery

u e

K.

0\0i

Tkeu'

K is

þrecomþøct

in

Lþ(Q)'

3.

Convergence

o1 the numerieal

solution.

Existence of the

exaet

solution. îhe

"lemmas

of the previous

section show

that the family

of numerical solutions

U

(depending

W!(Qr) norm, i.e. the

scheme. ^(1.5 ove have shown

that it is

stable

in We

are going now

to

^{discuss the}

and ^prove that the limit is

the

mention

that,

though our argumettts refer

to

subsequences of. approximate, solutions,

théy

reniain

valid-for the

^whole

of the

seçluence,

in

view

of

the uniquenss

of the exact

solution.

According

to the

maximnm.

principle

given

in

Iremma

2.1,

we have

o<Ú<¡z, o<u'(MonÇ,

if

U is extended

in

R,,\õ,,, by any valües which ^do not exceede

M.

Consequen- 11y,

botlt

farnilies

of"fuicîions

are bounded

in

Lo(Q),

I _S

_ø

_S f

^{oo' Hcnce}

iti"r" is a

subsequence

of

steps

{ft,} C _Ur}

^so

that

^U nn as

well as Uí,

convefge weakly

to 1 ^e

^Lr(Q),

^say.

^This

^limit

^is common as

it

was shown

in

[B]"(see

alsó t5l, I,"*tìrä ^4.3). In the

sequel we

shall rvrite, for

bre-

vlty, U,

insted

of

^Unn

Using the

^discrete

variational formulation of problem (1.5)-(l'7), we

have

þroved [5] ^the

^{following theorems:}

N

I

K-1

ON THE NUMERICAL ^SOLUTION 97

96, ^{ÉRVIN SCHECTITËÈ B}

I

TrrEoRÞM 3.1. Suþþose that U

rQ, -' ^{R is}

^{the solution}

of (l'5)-(t'7)^

ønd,

that

the subsequeôce u u

is

^{ckoseú,' as above.} Assume tka't cond'itions of ^' Lem'mø

2.3

^øre

fulfilled"

^Then

for ^k -

^{0 :}

(i)

q(Úr)

-

^q(X)

ⁱⁿ L(Q'), and a.e.

orl

.Q',

VO'C C

^Q

(ii)

_e(Ûo)

* _ç(x) in

LL(Q'),

VO' C C

^O

(iii)

_e(Ûn),.

- +*, ⁱ : r, 2, in

^{LL(Q',), Q',}

Vcc

^O.

Here

Ç' : O' X ]0¡ T[, -

designates

the

weak convergence'

while C C

means

the strict

inclusion.

THE9REM 3.2. Assume tkat cond.itions of tke þreuious ^theorem holil. Then

(i)

À

e L*(Q);

(ä) t

satisJí'øs (1.4)'.

From

Theorem

3.1 it follows llnat 09(a)/xr - ^L'(8), i:1,2, ^T!

^is

also clear that y ) 0 on Q ^{and that}

id.åâtisÎies

condition

^{(1.2); since}

U t,

*

_{X a,e.}

on

_Ç'.

In

order

to

show (ii)

it

suffices

to write the

difference problem

(1.5)-

(1.7)

in the variational form

(see

[5], ^(5'1)l:

I 40, -

ip*,(rt)$,,

-

õ.^(u)Ç,")d,xdt

I

a

+lã"@)Ç(x, 0)dx: ^0, _Ü ^{e D([0,} Îi x o)

o,) and pass

to the limit.

THEoREM

3.3.

Suþþose

,tkøt w is

the solwtion

of

_þroblern

'(1.1)-(1.3).

Then _9(-[J)u

-

^e(u)

^{in L(Q.}

^{Tke sarne}

^is

^t.rue

[* ^Uí. _-

-

piòoi."Co"Jibét

r

s"qüêo"" of subomains Q¡

as in

l,emma

2.5.

Suppose

K:{q(Ûn)}u

(or e(Uí)),

t

bounded

Á WI(Q)

and

hand by

Lemma

2.1 K on any point of

_Q. Thus completes

the

proof.

Rem ark.

Sinae the imbed'd'hing

wi(Q) -

^Lo(Q)

is

cornþact, accord.,ing to th.e Reltich-Kond,røshou ^theorem,

for

^any

^q

^{suck thøt'}

l<q<2

(in our particular

câse

r,: 2)

the conaergenae

from

^{the øbove theorem tøhes}

þlace

in

øny Lq(Q),

I

S ^q

<2.

REFDRENCES

Press, ^1975'

"ttnJti'

^iopics

in

^{Numerical Anaysls}

IrI'

6.

true Problems

of

the Møthematical ^Physics'

on ^d'es þroblèrnes ø'utt limí'tes non li'néøítes'

Exbticit Difference Aþþrorimalàon to Solue the

¿¿'íïi "püíiu¿i

a ^E quàii on s' Mathematica' vol'

RecelYetl 25, ll. lÐ75.

Føcullateo de Matemøticd' ø IJ nôaer sitã'li,i, B abe _ç -B olY ai'

Str. Kogdtrnàaeønø Nr. ¹

Cluj-Nøþoaø

?-Mathematica-Revued,analysenumériqueetdethéoriedel'apploximation_Tome8,N¡.1/1979