PARALEL PREDICTOR-CORRECTOR SCHEMES FOR PARABOLIC PROBLEMS ON GRAPHS

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1 COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol. 10(2010, No. 3, pp c 2010 Istitute of Matematics of te Natioal Academy of Scieces of Belarus PARALEL PREDICTOR-CORRECTOR SCHEMES FOR PARABOLIC PROBLEMS ON GRAPHS R.ČIEGIS1 AND N.TUMANOVA 1 Abstract We cosider a predictor-corrector type fiite differece sceme for solvig oe-dimesioal parabolic problems. Tis algoritm decouples computatios o differet subdomais ad tus ca be efficietly implemeted o parallel computers ad used to solve problems o grap structures. Te stability ad covergece of te discrete solutio is proved i te special eergy ad maximum orms. Te results of computatioal experimets are preseted Matematics Subject Classificatio: 65N20; 65M55. Keywords: parabolic problem, braced structures, fiite differece metod, predictor corrector algoritm, stability, covergece, parallel algoritms. 1. Itroductio May applied problems are described by reactio - diffusio equatios o braced structures. Te well-kow examples are give by euro simulatio models based o te Hodgki Huxley (HH reactio diffusio system [4]. I order to describe te fuctioig of te brai we must take ito accout te complex arcitecture of te syaptic coectios betwee euros, te spatial distributio of te membrae caels witi te braced structure of euros, ad te igly oliear electrical properties of te euros. Te amout of computatios eeded to make a matematical model fit experimetal data by explorig exaustively te parameter space grows expoetially wit te umber of parameters. Numerical algoritms for solvig HH type problems were proposed ad ivestigated i [1, 6, 7, 11]. Tey use te fiite differece metod for te approximatio of space derivatives ad apply differet teciques for umerical itegratio i time. I costructig umerical approximatios of reactio-diffusio systems o braced structures, very importat problem arises due to te approximatio of flux coservatio equatios at brac poits. Recetly, te predictor-corrector splittig ad domai decompositio metods ave bee widely used to solve elliptic ad parabolic PDEs i multidimesioal domais [3, 10, 16]. Importat results for domai decompositio metods are described i [5]. A geeral framework for te costructio ad aalysis of various classes of splittig ad domai decompositio algoritms is preseted i [13]. Very iterestig developmets of tese ideas are preseted i [15], were ew domai decompositio metods wit overlappig subdomais are ivestigated. Te mai idea is to coose a simple geeratig sceme for te classical approximatio of te cotiuous problem, te to icrease its stability by addig te regularizatio operator at te secod stage of developig a efficiet discrete sceme. Geeral operator stability coditios [12] ca be used to prove te stability of te ew discrete scemes tat are obtaied. 1 Vilius Gedimias Tecical Uiversity, Saulėtekio al. 11, Vilius, Lituaia. rc@vgtu.lt Uauteticated

2 276 R. Čiegis ad N. Tumaova Suc algoritms are well suited for parallel implemetatio. Similar teciques ca also be used for problems o grap structures (see [11, 14] for related discussios. Equatios o differet edges of te grap are decoupled ad ca be solved i parallel. Parallelizatio of te algoritm is very importat i solvig real-world problems we te amout of computatios icreases, ad te applicatio of parallel algoritms eables us to reduce substatially te CPU time. I tis paper, we cosider predictor ad predictor corrector type fiite differece scemes for solutig parabolic problems. Tese results ca be geeralized to problems o graps quite straigtforwardly. Te mai goal is to ivestigate te stability of te obtaied discrete algoritms wit respect to te approximatio errors itroduced at te predictor step of te algoritm. Stability aalysis i te maximum orm was performed i [2], but oly te coditioal stability ca be proved by tis metod ad o ifluece of te corrector step o te stability is obtaied. I te preset paper, we ave ivestigated te stability of te proposed discrete sceme i te eergy ad maximum orms. For ease of presetatio, we restrict ourselves to a detailed aalysis of oe-dimesioal liear parabolic problems. Te mai goal is to ivestigate te stability of te obtaied discrete algoritms wit respect to te approximatio errors itroduced at te predictor step of te algoritm. Te rest of te paper is orgaized as follows. Te matematical model of a liear parabolic problem ad te predictor-corrector algoritm are preseted i Sectio 2. Te covergece of te discrete solutio i te eergy ad maximum orms is ivestigated i Sectio 3. Some umerical results are preseted i Sectio 4, tey cofirm te obtaied teoretical results. Fially, some coclusios are give i Sectio Problem formulatio We cosider a oe-dimesioal liear parabolic problem u t = ( a(x u x x 0 < a 0 a(x a M, q(x 0, u(x,0 = u 0 (x, 0 x 1, q(xu(x,t+f(x,t, 0 < x < 1, t > 0, (2.1 u(0,t = µ L (t, u(1,t = µ R (t, 0 < t T Predictor corrector discrete sceme We costruct uiform grids for te x ad t variables, respectively, ω = {x j : x j = j, j = 1,...,N 1}, = 1/N, ω = ω {x 0,x N }, ω τ = {t : t = τ, = 0,...,N, Nτ = T}. O ω ω τ wedefietediscretefuctiou j = U(x j,t approximatigtesolutiou(x j,t at time t. Te backward time, te forward ad backward space differece quotiets wit respect to t ad x are defied by U t = U U 1, x Uj = U j+1 Uj τ, x Uj = U j Uj 1. Uauteticated

3 Parallel Predictor-corrector Scemes for Parabolic Problems o Graps 277 Let U, V be grid fuctios, te we defie te discrete ier product (U,V, te discrete L 2 orm U, ad te discrete H 1 semiorm U b by J 1 (U,V = U j V j, U = (U,U, U 2 b = j=1 J ( 2, b j 1/2 x U j j=1 were b(x is a give positive fuctio b(x > 0, x [0,1]. Recetly, te predictor-corrector splittig ad domai decompositio metods ave bee widely used to solve elliptic ad parabolic PDEs i multidimesioal regios ad o grap structures (see [11, 14] for related discussios. Te preseted algoritm is based o te stadard decompositio of te problem ito simple 1D problems o eac edge. New values of te solutio at brac poits are predicted by usig a explicit algoritm. I order to improve te stability of te domai decompositio algoritms, a ew approac was proposed i [9]. Te mai idea is to drop te values of te solutio at te iterface of two domais computed by te predictor algoritm ad compute ew values by usig te basic implicit discrete algoritm (correctio step. We ote tat predictor-corrector splittig fits well wit te geeral framework from [15], sice te predictor step implemeted as te explicit Euler algoritm ca be take as a geeratig sceme, ad te implicit Euler sceme at te corrector step ca be regarded as a regularized algoritm i tis framework. Problem (2.1 is approximated by te followig sceme. First, te domai Ω = [0,1] is divided ito K subdomais Ω = K Ω k, Ωk = [l k 1,l k ], l 0 = 0, l K = 1. I accordace wit tis domai decompositio, te space grid is decomposed ito subgrids ω = K ω k, were ω k = {x j : x j = j, j = j k 1,...,j k }, x = l k. Predictor step. First, we compute i parallel te ew values of te solutio at te boudary poits of te subdomais. Te explicit-implicit Euler approximatio is used to discretize te differetial equatio (2.1: Ũ j k U 1 j k τ = x ( a x U 1 j q k Ũj k +f 1 j k, k = 1,...,K 1. (2.2 Domai decompositio step. Secod, te solutios o eac subdomai are computed i parallel usig te implicit fiite differece sceme (2.3. Te predicted values Ũ j k are used as te iterface boudary coditios. ( U t = x aj+ 1 2 x Uj qj Uj +fj, x j ω, k k = 1,...,K 1, (2.3 U j k 1 = Ũ j k 1, U j k = Ũ j k, U 0 = µ L (t, U J = µ R (t. Uauteticated

4 278 R. Čiegis ad N. Tumaova Corrector step. Tird, usig te implicit fiite differece sceme (2.3 ad takig te solutio U computed at te secod step, we update i parallel te values of te solutio at te boudary poits of te subdomais Uj k, t = x( a + 1 x Uj q Uj k +fj k, k = 1,...,K 1. (2.4 Our goal is to ivestigate te stability ad te accuracy of te discrete algoritm. 3. Covergece aalysis Let us deote te error fuctios of te discrete solutio as Zj = Uj u(x j,t, Z j = Ũj u(x j,t, x j ω. By puttig tem ito te fiite-differece sceme we get a discrete problem for te error fuctios Zj, t = x( aj+ 1 2 x Zj qj Z j +ψ j, x j ω \{x ±1, k = 1,...,K 1}, (3.1 Zj k ±1, t = x( a ± x Zj k ±1 q ±1Zj k ±1 +a ± 1 2 Z j k Z 1 j k τ Z j k Z j k 2 +ψ j k ±1, (3.2 ( = x a + 1 x Z 1 j q Z ++ϕ j k, k = 1,...,K 1, (3.3 Z 0 = 0, Z J = 0, Z 0 j = 0, j = 0,...,J, (3.4 were ψ ad ϕ are te trucatio errors of te implicit ad explicit-implicit Euler scemes ψj = u + x( ( j, t aj+ 1 2 x uj qj u j +fj, ϕ j = u j, t + x aj+ 1 x u 1 j qj u j +f 1 j. 2 Let us assume tat te iitial data, te coefficiets, ad te exact solutio of problem (2.1 are smoot eoug. Usig te Taylor formula, we ca eatimate te trucatio errors of te fiite differece sceme (2.2 (2.4 as ψ j C(τ + 2, 1 j J, ϕ j k ψ j k Cτ, 1 k < K. (3.5 Next, we ivestigate te stability of te discrete solutio of te fiite differece sceme (2.2 ( Stability i H 1 Ourstabilityaalysisusesteresultsof[14]. Letā = (a 1/2+a +1/2/2, k = 1,...,K 1. Lemma 3.1. Let Z satisfy problem (3.1 (3.4. Te E E 0 +τ l=1 { K ψl 2 + [ 2τ olds, were te fuctioal E is defied as E = Z 2 a + qz 2 + 2τ2 ( a ā (ψl j k M (K 1τ + (ϕ l a 0 2 j ψj l 2 k ]}, (3.6 K 1 ā 1+τq ( x (a 1/2 x Z j k 2. Uauteticated

5 Parallel Predictor-corrector Scemes for Parabolic Problems o Graps 279 Proof. Multiplyig equatios (3.1 (3.2, respectively, by 2τ tz j, j = 1,...,J 1 ad addig te resultig equalities, we get 2τ tz 2 = 2τ ( ( tz, x aj 1 Z 2 x j 2τ + K 1 ( Z j k Zj k (a 1 tz j 1 +a + 2 tz 1 j k +1 2τ ( qz, tz +2τ ( ψ, tz. (3.7 Applyig equality 2Z = Z +Z 1 +τ tz ad te well-kow formula of summatio by parts, we obtai 2τ ( ( tz, x aj 1 Z 2 x j = Z 2 a + Z 1 2 a τ 2 tz 2 a, (3.8 2τ ( qz, tz = qz 2 qz 1 2 +τ 2 q tz 2. (3.9 Usig te Scwarz iequality ad te ε iequality, we ave 2τ ( ψ, tz 2τ tz 2 + τ 2 ψ 2. (3.10 It remais to estimate te error itroduced at te predictor step. From (3.1, (3.3 it follows tat We ote tat (1+τq ( Z j k Z j k = τ 2 t x ( a 1 2 x Z j k +τ(ϕ ψ j k. ( a 1 tz j 1 +a + 1 tz j +1 = 2 t x a 2 x 1 Zj +2ā k tz j k ( ( ( = 2 t x a 1 x Zj +2ā x a 1 x Zj +ψ. Applyig te Scwarz iequality, te ε iequality, ad te a priori estimate q j 0, we get te followig estimates: T 1 = 2τ3 ( ( t x a 1+τq 1 x Zj 2, T 2 = 4τ3 ā (1+τq ( ( x a 2 x 1 Zj k t x a 1 x Zj = 2τ2 ā (1+τq ( ( x ( a 2 x 1 Zj 2 ( ( k x a 1 x Z ( 1 2 ( j +τ 2 t x a 1 x Zj 2, T 3 = 4τ3 ā (1+τq t ( x a 1 x Zj ψ 2τ4 ā [ ( ] t x a (1+τq 1 x Zj 2 2τ 2 ā ( + ψ 2, T 4 = 2τ2 ( t x a 1+τq 1 x Zj (ϕ ψj k 2τ3 ( ( t x a 1+τq 1 x Zj 2 τ( + ϕ 2. 2 ψj k Now we will estimate te last term i a differet way ta it was doe i [14]. Te followig imbeddig teorem Z j 1 2 a 0 Z a is used. Te we obtai T 5 = 4τ2 ā (1+τq (ϕ j k ψj k tz j k 4τ2 a M ϕ j k ψ j k tz j k τ2 K 1 tz 2 a + a2 M (K 1τ2( ϕ 2. a 0 2 ψj k Uauteticated

6 280 R. Čiegis ad N. Tumaova Substitutig (3.8 (3.10 ito (3.7 ad usig te resultig iequalities T m, m = 1,...,5, we obtai E E 1 +τ{ 1 2 ψ 2 + K 1 [ 2τ Repeated applicatio yields te desired result (3.6. ( a ā (ψ j k M (K 1τ + (ϕ a 0 2 j ψj 2 k ]}. Remark 3.1. It follows from Lemma 3.1 tat te fiite differece sceme (2.2 (2.4 is ucoditioally A-stable. Note tat i [14] oly te ρ-stability wit ρ = 1+cτ was proved. Usig te stability estimate give i Lemma 3.1 ad te estimates of te trucatio errors (3.5, we get te followig covergece result i te H 1 orm. Teorem 1. Let U be te solutio of te fiite differece sceme (2.2 (2.4 ad u(x, t be te solutio of te differetial problem (2.1. Te 3.2. Stability i L U u a Ct ( τ τ τ. (3.11 I tis sectio, we ivestigate te asymptotic optimality of te error estimate (3.11. For simplicity of otatios, we restrict ourselves to te case K = 1 ad assume tat a(x 1, q(x = 0. Let us deote m = j 1 ad defie g = max 1 N max x j ω ψ j, g = max 1 N ϕ m ψ m, G = g +g. Te we cosider te statioary aalogue of te predictor-corrector algoritm x x W j = g, 1 j < m, W 0 = 0, Wm p = W m, x x W j = g, m < j < J, Wm p = W m, W J = 0, (3.12 W m W m τ = x x W m +G, x x W m = g, j = m. Te solutio of (3.12 ca be writte i explicit form A x j + g x m 2 x j(1 x j, 0 j m, W j = A 1 x j + g 1 x m 2 x j(1 x j, m < j J, We see tat te fuctio W is bouded by W j C (τ τ2, A = 2τ(x m x 2 m /2 ad tis estimate predicts a better covergece rate ta it follows from te a priori error estimate (3.11. g. Uauteticated

7 Parallel Predictor-corrector Scemes for Parabolic Problems o Graps Numerical experimets We ave applied te developed fiite differece sceme (2.2 (2.4 to te model problem (2.1, were we take T = 1 ad te followig coefficiets a(x 1, q(x = 0. Te source term f is cose suc tat te exact solutio is give by u = exp(texp(2x. Te computatioal domai is decomposed ito four subdomais at poits l 1 = 0.25, l 2 = 0.5, l 3 = 0.8. I te umerical experimets we ave tested te covergece of te algoritm for differet values of ad τ. Te error of te solutio is preseted i te uiform orm Te results are give i Table 4.1. Z N = max 1 j<j UN j u(x j,t. Table 4.1. Error of te solutio of te predictor corrector sceme at T = 1 N τ = 0.02 τ = 0.01 τ = τ = τ = Te results sow tat te global error i most cases coverges as O(τ 2 /, i.e., te error itroduced at te predictio step domiates te total error. It follows from Teorem 1 tat te predictor corrector sceme is ucoditioally stable wit respect to te iitial coditio i te eergy orm, i.e., E for te test problem 3 E = Z 2 a ++ 2τ2 ( 2. x x Zj k E 1 for > 0, were 1500 = 0.01, tau= 0.01 = 0.005, tau=0.005 = 0.005, tau = t Fig Dyamics of te error Z i time Uauteticated

8 282 R. Čiegis ad N. Tumaova But i te case of oliear problems we are iterested to get error estimates also i te uiform orm. I Fig. 4.1, we plot te dyamics of te error Z i time, we zero iitial coditios are disturbed oly at oe grid poit ZJ/2 0 = 1 ad te source term f Coclusios Te predictor corrector algoritm splits te computatios for eac time step ito a set of discrete oe-dimesioal parabolic problems wic ca be solved efficietly by te stadard factorizatio algoritm. Tis leads to a good parallelism of te algoritm. It is wellkow tat te simple predictio strategy loses te ucoditioal stability of te backward Euler algoritm. By te eergy estimates metod we ave proved tat te predictor-corrector algoritm is ucoditioally stable. Te relatio betwee te space ad time steps is still required i order to get te covergece of te discrete solutio, sice te trucatio error itroduced at te predictio step is of order O(τ 2 /. Ackowledgmets. We are grateful to te aoymous reviewer for te costructive remarks wic eabled us to improve te paper. Refereces 1. R. Cegis, Ivestigatio of differece scemes for a class of models of excitability, Comput. Mats. Mat. Pys., 32 (1992, o. 6, pp R. Čiegis ad N. Tumaova, Fiite-differece scemes for parabolic problems o graps, Lit. Mat. Joural, 50 (2010, o. 2, pp M. Dryja ad X. Tu, A domai decompositio of parabolic problems, Numer. Mat., 107 (2007, pp A. Hodgki ad A. Huxley, A quatitative descriptio of membrae curret ad its applicatio to coductio ad excitatio i erve, J. Pysiol. Lodo, 117 (1952, pp Y. Laevsky, Domai decompositio metods for te solutio of two-dimesioal parabolic equatios, i: Variatioal-differece metods i problems of umerical aalysis., vol. 2, Comp. Cetr. Sib. Brac, USSR Acad. Sci., 1987, pp M. Mascagi, Te backward Euler metod for umerical simulatio of te Hodgki Huxley equatios of erve coductio, SIAM J. Numer. Aal., 27 (1990, pp M. Mascagi, A parallelizig algoritm for computig solutios to arbitrarily braced cable euro models, J. Neurosci. Metods, 36 (1991, pp T. Matew, Domai Decompositio Metods for te Numerical Solutio of Partial Differetial Equatios, Lecture Notes i Computatioal Sciece ad Egieerig 61. Berli: Spriger, H. Qia ad J. Zu, O te efficiet parallel algoritm for solvig time depedet partial differetial equatios, i: Proceedigs of te 1998 Iteratioal Coferece o Parallel ad Distributed Processig Teciques ad Applicatios, CSREA Press, Las Vegas, NV, 1998, pp A. Quarteroi ad A. Valli, Domai Decompositio Metods for Partial Differetial Equatios. Numerical Matematics ad Scietific Computatio, Oxford Uiversity Press, USA, M. Rempe ad D. Copp, A predictor-corrector algoritm for reactio diffusio equatios associated wit eural activity o braced structures, SIAM J. Sci. Comput., 28 (2006, o. 6, pp A. Samarskii, Te Teory of Differece Scemes, Marcel Dekker, Ic., New York Basel, A. Samarskii, P. Matus, ad P. Vabiscevic, Differece Scemes wit Operator Factors, Matematics ad its Applicatios (Dordrect Dordrect: Kluwer Academic Publisers, H. Si ad H. Liao, Ucoditioal stability of corrected explicit-implicit domai decompositio algoritms for parallel approximatio of eat equatios, SIAM J. Numer. Aal., 44 (2006, o. 4, pp P. Vabiscevic, Domai decompositio metods wit overlappig subdomais for te time-depedet problems of matematical pysics, CMAM, 8 (2008, o. 4, pp Y. Zuag ad X. Su, Stabilized explicit implicit domai decompositio metods for te umerical solutio of parabolic equatios, SIAM J. Sci. Comput., 24 (2002, o. 1, pp Uauteticated