A Linearized Finite Difference Scheme with Non-uniform Meshes for Semi-linear Parabolic Equation

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1 Jun Zou A Lnearzed Fnte Dfference Sceme wt Non-unform Meses for Sem-lnear Parabolc Equaton Jun Zou Yangtze Normal Unversty Scool of Matematcs and Computer Scence Congqng Cna flzjzklm@6.com Abstract: In te present paper, a lnearzed dfference sceme wt non-unform meses for sem-lnear parabolc equaton s proposed. Te sceme s constructed accordng to te cange rule of te soluton by travellng wave soluton teory for partal dfferental equaton. Te exstence and unqueness of te numercal soluton are derved by lnear systems teory, and te convergence and stablty of te dfference sceme are proved by te dscrete energy metod. Numercal smulatons verfy te teoretcal analyss, te results sow tat te numercal soluton wt non-unform mess s more accurate tan tat wt unform meses n te sense of not costng muc more computng tme. It s concluded tat our sceme s effectve. Key Words: sem-lnear parabolc equaton, non-unform meses, dfference scemes, convergence, stablty Introducton Te Sem-lnear Parabolc Equatons ave wde applcatons n cemcal reacton, neural conducton, bologcal competton and oter felds. Te studes on tese equatons ave been a ot topc n past decades. It s of sgnfcance to explore teoretcally and numercally te solutons to tese equatons. Tere are many avalable works contrbuted to nvestgaton n ts feld for nstance see [-3]. Ames n [4] gave a large collecton of pyscal problems avng nonlnear parabolc equatons as models. Also te survey lsts varous metods for exact, approxmate and numercal solutons for tose examples. Based on tese equatons, tere ad been some fnte dfference metods suc as alternatng drecton teratve sceme, predctor-correctors metods and te lnearzed two or tree level dfference scemes [5-8]. Ramos n [9- ] compared varous fnte dfference scemes tat nclude explct, mplct and lnearzed scemes. Besdes tese fnte dfference scemes, Tang n [] studed fnte element metod of a nonlnear dffuson system. All metods mentoned above ave not taken rule cange of te analytcal soluton nto account; te rule s tat t canges quckly n some area, and slowly n oter area. In fact, ts rule can be deducted by travellng wave soluton teory for partal dfferental equaton. Accordng to ts rule, te tradtonal metods gven above ad a dsadvantage. Wen te exactness of numercal soluton s requred, one as to refne grd by ncreasng grd ponts. Ts way causes ncrease of computng amount. To overcome te drawback, te fnte dfference scemes wt non-unform meses ave attracted great attenton. Mattej and Smooke, Samarskj and s co-operators nvestgated te stablty and convergence of varable step space and tme algortms n te soluton of te mxed ntal-boundary problem of one-dmensonal parabolc equaton t = u fx, t x and two-dmensonal parabolc equaton t = u x u x fx, x, t n [3-5] respectvely. For te generalzed non-lnear parabolc systems u t = Ax, t, uu xx fx, t, u, u x, Zou constructed te general fnte dfference sceme wt non-unform meses and proved te exstence and st order convergence n L -norm of te dscrete solutons for te dfference sceme by te fxed pont tecnque n [6]. Yuan proved te unque solvablty and stablty for te dfference sceme constructed n [6] by te energy metod n [7, 8]. Ter work solved some unexpected penomenon, but ter proof s very complex.meanwle, tey ad no numercal experments to justfy ter teoretcal analyss. In order to solve te exsted problem, Zou and Hu constructed an mplct dfference scemes wt nonunform meses for te flame equaton, and tey prove E-ISSN: Issue, Volume, January 03

2 Jun Zou te unqueness, exstence, convergence and stablty of dfference soluton of te mplct sceme n [, ]. Te te sceme wt non-unform meses for space was constructed by a functon transformaton, but te mess for tme s stll unform. Te numercal experments were carred out to justfy tat te convergence of te soluton s st-order for tme. Tese results concde wt te prevous teoretcal analyss. However, n order to get te soluton to te mplct sceme, te teraton metod for non-lnear equatons needs to be appled, wc costs a quantty of tme for every tme step. To overcome ts drawback, we constructed a knd of lnearzed fnte dfference sceme on te base of mplct sceme n [, ] In ts paper, we wll nvestgate a lnearzed dfference sceme wt non-unform meses approxmatng to te followng Drclet problem of a semlnear parabolc equaton: t = u x fu, ux, 0 = φx, x [a, b], ua, t = αt, ub, t = βt, t [0, T ], 3 Let Ω T = [a, b] [0, T ]. In order to prove some propertes of te fnte dfference sceme, we mpose te followng condtons: I Te analytcal soluton of problem satsfes ux, t C 4,3,and tere exsts a postve constant C 0 satsfyng u u t u x u xx C 0. II Let fs be a two tmes contnuous dfferentable functon for s, tere exst two postve constants C and P satsfyng { fs, f s, f s } C wen s C 0 P. III Boundary value functons αt, βt are contnuous dfferentable functon for t, ntal functon φx s also contnuous dfferentable functon wt respect to x, and we ave α0 = φa, β0 = φb. Dfference Sceme and Notatons Let us dvde te rectangular doman Ω T = [a, b] [0, T ] nto te small rectangular grds { } Ω a = x0 < x =,..., < x I < x I = b,. 0 = t 0 < t <..., < t K = T Te t doman on space s [x, x ]. Te messteps of space s = x x wc are assumed to be unequal. Te tradtonal unform meses are appled for tme 0 = t 0 < t < t... t K < t K = T, = t k t k. We denote te maxmum value of space messteps s = max { 0 I }, te mnmum of space messteps s = mn { 0 I }, te rato of maxmum and mnmum of space messteps R =. So we denote dscrete functons u = {u k = 0,,,..., I, k = 0,,,..., K} on Ω T. Te oter notatons are as follows: t u k = uk u k δ xu k =, δ x u k = u k u k, δ x u k δ x u k, =, I u = max { u }, u = u, 0 I I δ x u = =0 δ x u, δ x u = max δu 0 I, By Taylor expanson, we construct te lnearzed dfference sceme: f u k tu k = δ xu k fu k, I, 0 k K, 4 u 0 = φx, 0 I, 5 u k 0 = αt k, u k I = βt k, 0 k K, 6 3 Exstence and Unqueness Teorem Tere exsts unque dfference soluton u k, =,,..., I, k =,,..., K satsfyng te dfference sceme 4-6 E-ISSN: Issue, Volume, January 03

3 Jun Zou Proof: Expandng te dfference sceme4-6, we get uk f u k u k uk = f u k u k fu k, I, 0 k K, Obvously, t s trangle lnear systems. Let a = b = c =, f u k,, for =,,..., I. By u k C 0 P Ts result can be gotten by matematcal nducton n secton 3 and assumpton II,wen tme step < C, we get f u k > C > 0. Fetcng te absolute value of b and c, b = c = f u k, = 3, 3 we get b > c. Smlarly we get b I > a I. For =, 3,..., I, summng up te absolute value of a, c, we get a c = = = Comparng t wt b and by f u k > 0, we get b > a c., It mples tat te coeffcent matrx s te strctly row dagonally domnant. Terefore, tere are unque soluton u k, =,,..., I, k =,,..., K to te lnear systems [3]. It mples tat tere exsts a unque numercal soluton to satsfy te dfference sceme Convergence In order to prove te convergence and stablty of te soluton to te dfference sceme 4-6, we mport four lemmas [6]. Lemma For any u = {u = 0,,,..., I} and v = {v = 0,,,..., I}, tere are I I =0 u v v = I v u u u 0 v 0 u I v I, u δv u 0 δv δv. u I δv I = I =0 δu δv Lemma 3 For any u = {u = 0,,,..., I} defned on te grd ponts {x = 0,,,..., I} wt unequal messteps, tere are relatons: u u, δu δu. Lemma 4 For any u = {u = 0,,,..., I} defned on te grd ponts {x = 0,,,..., I},tere are relatons: u l δu l u 0, ere 0 = x 0 < x <... < x I < x I = l. Teorem 5 For any u = {u = 0,,,..., I} defned on te grd ponts{x = 0,,,..., I},tere are relatons: u b δu b a u 0, ere a = x 0 < x <... < x I < x I = b. Remark: Te proof of Teorem 5 s completely same as tat of lemma 3 n [6].Ts result s only popularzed from [0,l] to general doman [a, b]. Lemma 6 Suppose te dscrete functon u = {u k k = 0,,,..., K} defned on te grd E-ISSN: Issue, Volume, January 03

4 Jun Zou ponts{t k k = 0,,,..., K} wt unequal messteps = { k = t k t k > 0 k = 0,,,..., K } satsfes recurrng relaton u k u k A k u k u k C k ten tere s u k e 3Atk u 0 Ct k e 3Atk, were messteps 0 = t 0 < t <... < t K < t K = T are suffcently small tat A, A and C are constants. Te convergence teorem and ts proof are as follows: max 0 k K ek, K =0 max 0 k K ek, δ xe k, = O, and max δ xe k 0 k K ux, t k. max δ xe k, 0 k K K ek e k =0 = O,, were U k = Proof: By Taylor expanson at pont x, t k,we get ux, t k = ux, t k t x, t k u t x, η k, were η k s between t k and t k. It mples tat t x, t k = ux, t k ux, t k 7 u t x, η k, 8 From te notaton, we get were t x, t k = t U k R k, 9 R k = u t x, η k. For te dffuson part, we ave ux, t k = ux, t k x x, t k u x x, t k 3 u x 3 ζ k, t k 3, 0 ux, t k = ux, t k x x, t k u x x, t k 3 u x 3 ζ k, t k 3, were ζ k s between x and x, ζ k s between x and x. respec- Multplyng 0, by, tvely, we get ux, t k = ux, t k Teorem 7 Suppose tat te ntal boundary problem of partal dfference equatons -3 satsfy assumptons I, II and III, te messteps,, be suffcently small and R be boundary. We denote te error of dscrete soluton e = {e k = U k uk = 0,,,..., I; k = 0,,,..., K},ten tere are estmates x x, t k u x x, t k 3 3 u x 3 ζ k, t k, ux, t k = ux, t k x x, t k u x x, t k 3 3 u x 3 ζ k, t k, 3 Addng to 3, ten dvdng by, we get ux, t k = u x x, t k ux, t k ux, t k 3 u x 3 ζ k, t k, 3 u x 3 ζ k, t k 4 E-ISSN: Issue, Volume, January 03

5 Jun Zou From te notaton, we get were u x x, t k = δ xu k R k, 5 R k = 3 u x 3 ζ k, t k 3 u x 3 ζ k, t k. For te reacton part, smlarly by Taylor expanson, we ave fux, t k = fux, t k f ux, t k t x, t k { f ux, η k t x, η k f ux, η k u t x, η k }, were η k s between t k and t k. From te notaton,we get 6 fux, t k = fu k f U k t U k R 3 k, 7 were R 3 k = { f ux, η k t x, η k f ux, η k u t x, η k }. Substtutng 9,5 and 7 nto ntal problem of Eqs. -3, we get f U k tu k = δxu k fu k R k, I, 0 k K, 8 U 0 = φx, 0 I, 9 U k 0 = αt k, U k I = βt k, 0 k K, 0 were R k = R k R k R3 k = O, so we get te dfference sceme wt non-unform meses 4-6. Subtractng 4-6 from 8-0, we arrve at te error equatons: f U k te k = δ xe k fu k fuk [f U k f u k ] tu k R k, I, 0 k K, e 0 = 0, 0 I, e k 0 = 0, ek I = 0, 0 k K, 3 By te dfferentablty of f and te dfferental mean value teorem, te second and te trd term of are canged nto fu k fu k = f ξ k ek, 4 were ξ k s between U k and u k. Te fourt term of s canged to [f U k f u k ] tu k = f ξ k ek tu k f ξ k ek tu k f ξ k ek tu k = f ξ k ek tu k f ξ k ek te k, 5 were ξ k s between U k and u k. Substtutng 3-4 to 0,we get f U k te k = δ xe k f ξ k ek f ξ k ek tu k f ξ k ek te k R k, I, 0 k K. 6 By assumpton II, we know f U k C. It mples wen < C, Set we ave C f U k C, C C = f U k. C C, C 3 = C 3 Terefore by 6, we get t e k = C, f U k C. 7 f U k δxe k f ξ k ek f ξ k ek t U k f ξ k ek t e k R k. 8 E-ISSN: Issue, Volume, January 03

6 Jun Zou Multplyng 8 by δ ek from to I, we get I I = δxe k e k e k, and summng up f U k xe k δ δxe k f ξ k ek f ξ k ek tu k f ξ k ek t e k R k, 9 By lemma and te defnton of norm, te left and of 9 can be wrtten as I δ xe k e k e k = δ xe k δ xe k δ xe k e k. In fact, te left and of 9 s as follows I I = δxe k δxe k δxe k e k e k I = I II, δxe k e k e k e k δxe k e k e k 30 By te defnton of te second order dfference quotent, lemma and te defnton of te frst order dfference quotent n -norm,we get I I = δ x e k e k δ x e k e k = δ x e k e k, Smlarly, we get I II = δ x e k δ x e k e k = = I I δ x e k δ x e k δ x e k [ δ x e k δ x e k δ x e k δ x e k δ x e k δ x e k δ x e k ] = = I δ x e k δ x e k I [ δ x e k δ x e k δ x e k ] I ] δ x e k [ δ x e k δ x e k δ x e k δ xe k δ xe k = δ xe k e k δ xe k δ xe k, By -norm s defnton of two order dfference dvded and nequalty 7, te frst term of rgt and n 8 satsfes I f U k δ xe k δ xe k, By 9,30 and 3, we get C 3 δ xe k, 3 I C 3 δxe k f U k xe k δ f ξ k ek f ξ k ek tu k f ξ k ek te k R k δ xe k δ xe k δ xe k e k δ xe k δ xe k. By transposng, we get δ xe k δ xe k C 3 δ xe k I f U k xe k δ f ξ k ek f ξ k ek tu k f ξ k ek t e k R k, 3 E-ISSN: Issue, Volume, January 03

7 Jun Zou Usng Young s nequalty and nequalty 7, te rgt and of 3 s canged to I f U k xe k δ f ξ k ek f ξ k ek t U k f ξ k ek t e k R k ε C 3 δxe k C I ε C 3 f ξ k ek f ξ k ek t U k f ξ k ek t e k R k, 33 From te second term of te rgt and n 33, we get: I C ε f ξ k C ek f ξ k ek t U k 3 f ξ k ek t e k R k 8C ε C 3 I I { I f ξ k ek f ξ k ek t U k f ξ k ek t e k I = Q Q Q 3 Q 4, were Q = 8C ε C 3 Q = 8C ε C 3 Q 3 = 8C ε C 3 I I I Q 4 = 8C I ε R C k. 3 f ξ k ek, } Rk f ξ k ek t U k, f ξ k ek t e k, We now estmate Q =,, 3, 4 as follows. By, snce e 0 = 0, u 0 max a x b ϕx C 0, 34 by nducton ypotess, tere exst postve constant P satsfyng max 0 l k el P. Terefore, we ave e l P, u l U P C 0 P [7] wen l = 0,,,..., k. Usng assumpton II, we get and Q 8C C ε C 3 Q = 8C ε C 3 8C C ε C 3 I I 3C 0 C C ε C 3 I e k f ξ k ek = 8C C ε C 3 e k, U k U k e k U k U k I = 3C 0 C C ε C 3 e k, Q 3 = 8C ε C 3 I 6C C P ε C 3 e k f ξ k ek I e k e k e k e k 6C C P ε e k e k C, 3 Q 4 C 4 O, were C 4 depends upon C and rato constant R of messteps. Let ε =. From 3, 34 and te nequaltes about Q, Q, Q 3, Q 4, we get δ x e k δ x e k C 3 δ xe k C 5 e k ek O C 5 e k e k δ x e k δ x e k O, 35 E-ISSN: Issue, Volume, January 03

8 Jun Zou By lemma 4, formula 35 can be wrtten as δ x e k δ xe k C 6 e k ek O. By lemma 5, we get max δ xe k C 7, 0 k I were C 7 depends on C and te rato constant R of messteps. Terefore, max 0 k K ek, K =0 δxe k, O, So, we ave By lemma 3,we ave Wen we ave max 0 k K ek, K =0 max δ xe k, 0 k K ek max 0 k K0 I ek O. e k max δ xe k 0 k K 0 I δ x e k C 7 R. 5 Stablty s suffcently small and R s boundary, max δ xe k = O,. 0 k K In order to prove stablty of te dfference sceme, we mport te ntal boundary problem v t = v x fv ωx, t, 36 vx, 0 = φx ψx, x [a, b], 37 va, t = αt, vb, t = βt, t [0, T ], 38 were ωx, t, ψx s smoot enoug. Problem ave unque soluton vx, t, wc satsfy te assumptons I,II and III. Suppose tat v k, = 0,,..., I, k = 0,,..., K satsfy te followng dfference sceme: f v k t v k = δ xv k fv k ω k, I, 0 k K, 39 v 0 = φx ψ, 0 I, 40 v k 0 = αt k, v k I = βt k, 0 k K, 4 were ω k = ωx, t k, ψ = ψx,so we ave te followng stablty teorem. Teorem 8 Suppose u k s te numercal soluton of te dfference sceme 4-6,v k s te numercal soluton of te dfference sceme 36-38,denote z k = v k uk, ten wen, s suffcently small, s suffcently small too. Ten K δz k C ψ z k k=0 K k=0 ω k, were C doesn t depend on and wc s a constant. Proof: Subtractng 4-6 from 39-4 and by matematcal treatment, we get f v k t z k = δ xz k fv k fu k f v k f u k tu k ωk, I, 0 k K, 4 z 0 = ψ, 0 I, 43 z k 0 = 0, zk I = 0, 0 k K, 44 By te dfferentablty of f and te dfferental mean value teorem, te second and te trd term of 4 are canged to fv k fu k = f ξ 3 k zk, 45 were ξ 3 k s between vk and uk. Smlarly te fourt term of4 s canged to f v k f u k tu k = f ξ 4 k zk tu k, 46 were ξ 4 k s between vk and u k. Tus formula 4 becomes nto f v k t z k = δ xz k f ξ 3 k zk f ξ 4 k zk tu k ωk, I, 0 k K, 47 E-ISSN: Issue, Volume, January 03

9 Jun Zou Multplyng 47 by z k to I, we get I f v k z k z k z k = I I δxz k z k f ξ 4 k zk zk 0 k K,, and summng up from I I f ξ 3 k zk zk By proper deformaton, 48 s canged to I z k z k z k I = I I I f ξ 3 k zk zk f v kzk ω kzk, I I ω kzk, 48 δxz k z k f ξ 4 k zk zk f v kzk zk 0 k K, 49 By te metod n reference [, ], te left and n 49 s wrtten as I z k z k z k = I z k = z k z k z k I z k z k z k zk zk zk = zk z k zk z k, 50 By lemma and te defnton of -norm, te frst term of te rgt and n 49 s wrtten nto I δxz k z k = I δ x z k = δz k, 5 By assumpton II and te defnton of -norm, te fourt term of te rgt and n 49 as estmaton C I I f v kzk z k = C z k, 5 Usng te mean nequalty and te defnton of - norm, te second term, te trd term, te fft term and te sxt term ave estmatons as follows I f ξ 3 k zk zk I C z k I C z k = C z k C z k, I f ξ 4 k zk zk C z k I I ω kzk 53 C z k, 54 f v kzk zk C z k C z k, 55 I ω k I z k = ωk zk, Combnng 50-56, we ave z k zk δzk 56 C 8 z k zk ωk, 57 were C 8 s dependent on C, but ndependent of and. By dscrete Gronwall s nequalty and lemma, we ave z k K k=0 C 8 ψ K δz k k=0 Terefore, te Teorem 8 s proved. ω k. 58 E-ISSN: Issue, Volume, January 03

10 Jun Zou 6 Numercal Experments and Concluson Numercal example We apply te dfference sceme proposed n ts paper to te followng ntal boundary problem: unform mes alpa=3 alpa=0 t = u x u u, ux, 0 =, x [ 50, 50], x e u 50, t = e 5 t, u50, t =, t [0, 0], e 5 t te classcal soluton s ux, t =. e x t Frstly, we ntroduce te generaton metod of non-unform mess. From te curve of ntal functon see n Fgure, we can see tat te curve vary quckly near x = 0, but t canges gently near te two 50 snαξ endponts. Usng te transformaton x = snα as used n [], we transform te unform grd nodes ξ n [, ] to non-unform grd nodes x n [ 50, 50]. From Fgure, we see tat te grd nodes are centralzed near x = 0, te grd nodes are relatve sparse on te nterval endponts, te bgger te transformaton parameter α s, te more te grd nodes s centralzed Fgure : nonunform grd nodes canged under transformaton wt parameter α wen I = 00, = 0.5. Ts fgure only s an example. Te smlar results for oter grd partton can be obtaned by te same metod. Error s norm of Numercal Soluton Mes Transformaton Parameter ntal functon Fgure 3: Varaton of Error of te Numercal Soluton n L Norm ux, Fgure : te curve of te ntal functon Secondly, we searc te optmzed transformaton parameter α for dfferent grd partton. Here te optmal parameter s te parameter tat makes te numercal soluton s error attans t mnmum. In Fgure 3, numercal soluton s error s n te sense of L norm, x From Fgure 3, we see tat te error decays wt te ncreasng of te transformaton parameter α. It mples tat te more te grd nodes are centralzed, te less te error s. But ts grd centralzaton cannot be unlmted, ts s because R and may be very bg wen te grd nodes are centralzed to a certan extent. As a result, tey do not satsfy te condton of Teorem 7 and Teorem 8 n wc tese two values are te boundary. Te numercal oscllaton appearng n te rgt and of te curve can prove ts pont. So te centralzaton parameter α must be cosen exactly, tat develops te exactness of te dfference soluton and ensures te stablty and convergence of te numercal soluton. By ts metod, we get te optmal transformaton parameter α = 3.8. Smlarly, we can get te optmal transformaton parameter of te oter grd partton. E-ISSN: Issue, Volume, January 03

11 Jun Zou Table : Numercal results lnearzed unform lnearzed non-unform mplct non-unform I α tme E tme E tme E / / / / / / / / / Te trd s to get te mnmum parameter of te same space freedom degree and dfferent tme steps. Altoug te transformaton parameter α of dfferent grd partton can be appled to solve te numercal soluton and te errors are mnmum, te convergence order cannot be tested because of te dfferent parameters. In order to test te convergence order for, we get te mnmum parameter by comparng te dfferent optmzed parameters, wen space freedom degrees are same, tme steps are n alf n turn. Te parameters α are appled to numercal solvng wc can justfy te convergence order for. Te value of α and te computng results are lsted n Table. Te L norm of te errors for te cosen dfferent α are lsted n Table wen T = 0s, I = 00, I = 00 and I = 400. From Table, we can see tat te lnearzed dfference sceme wt nonunform meses put forward n ts paper s more accurate tan tat wt unform meses. Meanwle, t costs less computng tme tan mplct dfference sceme wt non-unform meses of [,] on condton tat te exactness of te numercal soluton as lttle dfference. In addton, It s known tat te convergence order of te numercal soluton s st order and stable wc s proved n Teorem 7 and Teorem 8. Summarly, te lnearzed dfference sceme studed n ts paper s effectve. Acknowledgements: Te researc was supported by Scence and Tecnology Researc Projects of Congqng Cty Educaton Commsson No. KJ308. References: [] L. Bledjan, Computaton of tme-depent lamnar fame strutrure, Comb. Flame, 0, 973, pp.5-7. [] M. E. Gurtn, On te dffuson of bologcal populaton, Matematcal Boscence, 33, 977, pp [3] R. Ftzug, Impulse and pysology states n models of nerve membranes, Bopys Journal, 96, pp [4] W. F. Ames, Nonlnear Partal Dfference Equatons n Engneerng, Academc Press, New York, 97, Vol.II, pp.0-8. [5] G. W. Yuan, L. J. Sen, Y. L. Zou, Uncondtonal stablty of parallel alternatng dfference scemes for sem-lnear parabolc systems, Appled Matematcs and Computaton, 7, 00, pp [6] D. A. Voss, A. Q. M. Kalq, A lnearly mplct predctor-correct metod for reacton-dffuson equatons, Computers and Matematcs wt Applcaton, 38, 999, pp [7] H. Wu, A Dfference Sceme for Sem-lnear Parabolc Equatons Regardng te Dstrbuted Control, Matematcal Applcata, 94, 006, pp In Cnese [8] Z. Sun, A Class of Second-Order Accurate Dfference Scemes for Quas-lnear Parabolc Equatons, Matematca Numerca Snca 64, 994, pp In Cnese [9] J. I. Ramos, Numercal Soluton of reactve- Dffusve Systems, Internatonal Journal of Computer Matematc, , pp [0] J. I. Ramos, Numercal Soluton of reactve- Dffusve Systems,,Internatonal Journal of Computer Matematcs 8, 985, pp.4-6. [] J. I. Ramos, Numercal Soluton of reactve- Dffusve Systems,,Internatonal Journal of Computer Matematcs 83, 985, pp [] S. M. Tang, S. D. Qn, R. O. Web, Nonlnear dffuson system s numercal soluton, Appled Matematcs and Mecancs, 8, 99, pp In Cnese [3] R. M. M. Mattej, M. D. Smooke, Stablty and convergence of lnear parabolc mxed ntalboundary value problems on nonunform grds, Appled Numercal Matematcs, 66, 990, pp [4] A. A. Samarskj, P. N. Vabscevc and P. P. Matus, Fnte-dfference approxmaton of ger accuracy order on nonunform grds, Dfferental Equatons, 3, 996, pp [5] A. A. Samarskj, V. I. Mazukn and P. P. Matus, Fnte-dfference sceme on nonunform grds for a two-dmensonal parabolc equaton,dfferental Equatons, 347, 998, pp [6] Y. L. Zou, Dfference scemes wt nonunform meses for nonlnear parabolc system, Journal of Computatonal Matematcs 44, 996, pp E-ISSN: Issue, Volume, January 03

12 Jun Zou [7] G. W. Yuan, Unqueness and stablty of dfference soluton wt nonunform meses for nonlnear parabolc systems, Matematca Numerca Snca,, 000, pp In Cnese [8] Y. L. Zou, L. J. Sen and G. W. Yuan, On te convergence of teratve dfference scemes wt nonunform meses for nonlnear parabolc systems, Advances n Matematcs, 53, 996, pp.8-8. [9] Y. L. Zou, L. J. Sen and G. W. Yuan, On some practcal dfference scemes wt ntrnsc parallelsm for nonlnear parabolc systems, Advances n Matematcs, 5, 996, pp.9-9. [0] Y. L.Zou, G. W. Yuan, Te general dfference scemes wt ntrnsc parallelsm for nonlnear parabolc systems, Scence n Cna Ser. A, 7, 997, pp.05-.in Cnese [] J. Zou, B.Hu, Researc on flame equaton s fnte dfference sceme wt non-unform mes, Journal of scuan unversty natural scence edton, 466, 009, pp in Cnese [] J. Zou, Y. Y. Yu, B. Hu, Unqueness and Stablty Study of a Knd of Fnte Dfference Sceme wt Non-Unform Mes, Journal of scuan unversty natural scence edton, to appear. In Cnese [3] R. L. Burden, J. D. Farer, Numercal Analyss, 7t ed. Prndle, Boston. 00. E-ISSN: Issue, Volume, January 03