Blanca Bujanda, Juan Carlos Jorge NEW EFFICIENT TIME INTEGRATORS FOR NON-LINEAR PARABOLIC PROBLEMS

Opuscula Matematica Vol. 26 No. 3 26 Blanca Bujanda, Juan Carlos Jorge NEW EFFICIENT TIME INTEGRATORS FOR NON-LINEAR PARABOLIC PROBLEMS Abstract. In tis work a new numerical metod is constructed for time-integrating multidimensional parabolic semilinear problems in a very efficient way. Te metod reaces te fourt order in time and it can be combined wit standard spatial discretizations of any order to obtain unconditinally convergent numerical algoritms. Te main teoretical results wic guarantee tis property are explained ere, as well as te metod caracteristics wic guarantee a very strong reduction of computational cost in comparison wit classical discretization metods. Keywords: fractional step metods, non-linear parabolic problems, convergence. Matematics Subject Classification: 65M6, 65M12. 1. INTRODUCTION In tis paper we deal wit te construction of numerical metods wic efficiently integrate te following non-linear parabolic problems: find y(t) : [t, T ] H, a solution of { y (t) = L(t)y(t) + f(t) + g(t, y(t)), y(t ) = y. Here H is a Hilbert space of functions defined on a certain domain Ω R n, L(t) is a linear second order elliptic differential operator, f(t) is te source term, g(t, y(t)) is a non-linear function wic meets some additional requirements and y y ( x) is te initial condition. Te numerical integration of tis problem can be viewed as a double process of discretization wic adequately combines a time integrator (typically an adapted ODE Solver) wit a discretization of te spatial variables. It is well known tat te use of classical implicit scemes for te numerical integration of tis type of problems poses mainly two problems. On te one and, te convergence of te numerical sceme is usually obtained by imposing excessively restrictive stability requirements wic force (1) 47

48 Blanca Bujanda, Juan Carlos Jorge us to use low order metods (like te implicit Euler rule) or complicated fully implicit metods for reacing iger orders of convergence in time. On te oter and, we must use slow iterative metods to resolve te internal stage equations because tey are defined as te solution of large non-linear systems. Also, in tis type of scemes, te convergence is deduced by imposing strong stability requirements (for example, B-stability). In [5] a semiexplicit metod, wic is used to determinate te contribution of te linear term, is combined in an additive way wit an explicit metod used for te contribution of te non-linear term. Tus, linear systems only appear in te computation of te internal stages. In [7], splitting metods are applied in te numerical integration of some non-linear advention-diffusion-reaction problems; ere te advention terms are integrated wit an explicit metod. In order to avoid tese problems, in [3] a new class of linearly implicit metods is developed; tey are built by combining a standard spatial discretization of Finite Differences or Finite Elements type wit a suitable time discretization, wic is deduced from a Fractional Step Runge Kutta (FSRK) metod to deal wit te linear terms and an explicit Runge Kutta metod to define te contribution of te non-linear term. By applying tis new linearly implicit metods (see [1, 2, 3]) we obtain numerical algoritms wic are convergent under a single requirement of te linear absolute stability type. It is well known (see [8]) tat te Fractional Step metods are very efficient in integrating multidimensional linear parabolic problems if we suitably decompose te linear elliptic operator and te source term; te advantages of tese metods manifest temselves also in tese new linearly implicit metods. Tus in te resolution of te internal stages only we must solve linear systems wit very simple matrices and it is not necessary to apply classical iterative metods to solve tem. For example, if te elliptic linear operator L(t) is n d i(t) 2 y x 2 i + n v i(t) y x i + n k i(t), and we realize a spatial discretization by applying classical Finite Differences, te matrices wic appear at eac stage are tridiagonal; consequently, te computational cost of te final metod is of te same order as tat of an explicit metod. 2. NUMERICAL SCHEME By applying firstly a spatial discretization process of type Finite Difference, Finite Elements,... to problem (1) we obtain a family of Initial Value Problems wic depend on te parameter (, ] used to discretize in space as follows: Find Y (t) : [t, T ] V, a solution of { Y (t) = L (t)y (t) + f (t) + g (t, Y (t)), Y (t ) = y. (2) For eac it is common to consider a finite dimensional space V ; suc space is te space of discrete functions on a mes in Finite Differences, te subspace of H of piecewise polynomial functions in a classical Finite Element discretization. We

New efficient time integrators for non-linear parabolic problems 49 suppose tat suc spatial semidiscretization process is uniformly convergent of order q, i.e., for sufficiently smoot functions y(t) tere is π y(t) Y (t) C q, t [t, T ]. (3) Here, te operator π denotes te restriction to te mes nodes if Finite Differences used and if we use te standard Finite Elements, ten π are suitable projections in te space V. In order to integrate efficiently tis problem by applying te linearly implicit numerical metods wic are first appeared in [3], we must adequately decompose te spatial discretization of te elliptic operator and te source term in n addends as follows: L (t) = n L i(t) and f (t) = n f i(t); te FSRK metod will act on tese terms. Te contribution of te non-linear term g (t, U (t)) is given by te explicit RK metod. From tis process, we obtain te following sceme: s Y m+1 = Y m +τ were i Y m,i = Y m+τ b ki i ( ) L ki(t m,i )Y m,i + f ki(t m,i ) + τ s ( ) i 1 a kj ij L kj(t m,j )Y m,j + f kj(t m,j ) + τ b n+1 i g (t m,i, Y m,i ), (4) a n+1 ij g (t m,j, Y m,j ). In tis algoritm, we denote wit Y m te numerical approximations to te exact solution y(t m ) at te times t m = t + m τ (τ is te time step), U = U (t ), k i, k j {1,, n}, n is te number of te levels of te metod, t m,i = t + (m + c i ) τ and te intermediate approximations U m,i for i = 1,, s are te internal stages of te metod. By setting some coefficients to zero in te latest formula as follows A k = (a k ij) were a k ij = b k = (b k j ) were b k j = a n+1 ij if k = n + 1 y i > j, a kj ij if k = k j and i j, oterwise, b n+1 j if k = n + 1, b kj j if k = k j, oterwise, we can organize tese coefficients in a table wic can be considered as an extension of te tables introduced by Butcer for standard RK metods, in te following way C e A 1 A 2... A n A n+1 (b 1 ) T (b 2 ) T... (b n ) T (b n+1 ) T were C = diag(c 1,, c s ) and e = (1,, 1).

41 Blanca Bujanda, Juan Carlos Jorge We will use (C, A i, b i ) n+1 to refer ro tis metod sortly and (C, Ai, b i ) to denote every standard RK metod involved in (4). To study te convergence of numerical sceme (4) we introduce, as usual, te global error at te time t = t m by E m = π y(t m ) Y m ; tus we say tat (4) is uniformly convergent of order p in time and of order q in space, if E m satisfies E m C( q + τ p ). (5) To get tis bound, we begin by studying te stability of numerical sceme (4); tus we rewrite it in a more compact way by using te following tensorial notation: given M (m ij ) R s s and v (v i ) R s, we denote M (m ij I V ) V s s and v (v i I V ) V s; we group te evaluations of te source terms f i(t) and of te linear operators L i (t) for i = 1,, n and for all m = 1, 2,, Fi m = (f i(t m,1 ),, f i (t m,s )) T V s and ˆL m i = diag (L i(t m,1 ),, L i (t m,s )) ( ) T L(V, V ). We also group te contribution of te stages Ỹ m = Y m,1,, Y m,s ( T V s and te nonlinear term Ĝm (Ỹ m) = g (t m,1, Y m,1 ),, g (t m,s, Y m,s )) V s. In tis way we obtain n n (Ī τ A i ˆL m m i )Ỹ = ē Y m + τ A i Fi m + τ A n+1 Ĝm (Ỹ m ), Y m+1 = Y m + τ n (b i ) T (ˆLm i Ỹ m + F m i ) + τ (b n+1 ) T Ĝ m (Ỹ m ). Wen te operator ( Ī n Aj ˆL j) m is invertible 1), te numerical solution can be written as were Y m+1 = R ( τ ˆL m 1,, τ ˆL m n) Y m + S ( τ ˆL m 1,, τ ˆL m n, τf m 1,, τf m n + T (τ ˆL m 1,, τ ˆL m n, τ Ĝm (Ỹ m )), (6) R(τ ˆL m 1,, τ ˆL m n) = Ī + n (b i ) T τ ˆL n 1ē, m i (Ī A j τ ˆL j) m (7) ) and S ( τ ˆL m 1,, τ ˆL m n, τ F1, m, τ Fn) m = (8) n ( = τ (b i ) T Fi m + ˆL m (Ī n i τ A j ˆL m ) 1 ( n j τ A k Fk m ) ) T (τ ˆL m 1,, τ ˆL m n, τ Ĝm (Ỹ m )) = (9) n = τ (b i ) T (Ī n ˆLm i A j ˆL m 1τA n+1 j) Ĝ m (Ỹ m ) + τ (b n+1 ) T Ĝ m (Ỹ m ). k=1 1) In [4] it is proved tat tis operator is invertible and tat its inverse operator is bounded independently of and τ.

New efficient time integrators for non-linear parabolic problems 411 Note tat first term (7) is te transition operator if we consider linear omogeneous problems; te contribution of te source term appears in second term (8) and, finally, te contribution of te nonlinear term g (t, Y (t)) is defined by tird addend (9). To study te stability of te numerical sceme we must bound tese tree addends (see [1]). Te contribution of te linear terms, given by (7), can be bounded by applying te following result (see [4]): Teorem 2.1. Let an A-stable FSRK metod be given suc tat all teir stages are implicit (i.e., n k=1 ak ii, for i = 1,, s) or its first stage only is explicit (i.e., n k=1 ak ii, for i = 2,, s and aki 11 = for all i = 1,, s) and it also satisfies (,,, 1) T A i = (b i ) T and a k1 ss, and let {L i (t)} n be a linear maximal coercive system of operators suc tat: a) for eac t [, T ] te system of operators {L i (t)} n is commutative, b) tere exist n constants M i suc tat L i (t )Y L i (t)y t t C L i (t)y t, t [t, T ]. Ten tere exists a constant γ, independent of τ, suc tat R( τ ˆLm 1,, τ ˆL m n) e γτ (1) olds. In [4], te following bound for second addend (8) is also proved to old: S( τ ˆLm 1,, τ ˆL m n, τ F m 1,, τ F m n) C τ n F m i To finis te study of te stability, we must bound tird term (9); in [1], te following result is proved, rewritten for our purposes: Teorem 2.2. Let (4) be a linearly implicit metod meeting te conditions of Teorem 2.1. If we use suc metod to integrate nonlinear parabolic problem (2) in time, were te nonlinear part g (t, Y ) satisfies g (t, X ) g (t, Y ) L X Y, t [t, T ], ten:. T (τ ˆL m 1,, τ ˆL m n, τ Ĝm ( X m )) T (τ ˆL m 1,, τ ˆL m n, τ Ĝm (Ỹ m )) C τ X m Y m. (11) To study te consistency of te numerical sceme, we define te local error of te time discretization metod as e m = Y (t m ) Ŷ m, were Ŷ m is obtained wit one step of (4) taking as starting value Y m 1 = Y (t m 1 ). We say tat (4) is uniformly consistent of order p if e m C τ p+1, m. (12) In [2] a teorem wic can be rewritten for tis case as follows is proved:

412 Blanca Bujanda, Juan Carlos Jorge Teorem 2.3. Let us assume tat problem (2) satisfies te smootness requirements Y (p+1) i 1 (t) C, L (ϱ1) i 1 (t) L(ϱ l 1) i l 1 (t)y (ϱ l) i l (t) C, t [t, T ], l {2,, p}, (i 1,, i l ) {1,, n + 1} l, (ϱ 1,, ϱ l 1 ) {,, p 1} l 1 and ϱ l {r + 1,, p} suc tat 2 l + l ϱ k p + 2, were Y i (t) = L i(t)y (t) + f i (t), i = 1,, n + 1 wit L (n+1) (t) = g Y (t, Y (t)) and f (n+1) (t) = g Y (t, Y (t)) Y (t) + g (t, Y (t)). Let (C, A i, b i ) n+1 be a linearly implicit FSRK metod satisfying te reductions (C) k e k A j (C) k 1 e =, j = 1,, n + 1, k = 1,, r wit r = E [ ] p 1 2 togeter wit te order conditions (b i1 ) T (C) ρ1 e = 1 ρ 1 + 1, Ten (12) is true. k=1 (b i1 ) T (C) ρ1 A i2 (C) ρ2 A i l (C) ρ l e = l 1 (l j + 1) + l {2,, p}, (i 1,, i l ) {1,, n + 1} l, (ρ 1,, ρ l 1 ) {,, p 1} l 1 and ρ l {r + 1,, p 1} suc tat 1 l +, l ρ k k=j l ρ k p. To obtain te convergence of te sceme (bound (5)), we decompose te global error as follows: k=1 E m π y(t m ) Y (t m ) + Y (t m ) Ŷ m + Ŷ m Y m were we ave used te intermediate approximations Y (t m ) and Ŷ m. Te space discretization is convergent of order q, tus by applying (3) we obtain E m C q + e m + Ŷ m Y m. Now we use te fact tat te time discretization is consistent of order p (i.e. it satisfies (12)) E m C q + C τ p+1 + Ŷ m Y m. Finally, we bound te tird addend it by using (6) (9) as follows Ŷ m Y m = R ( τ ˆL m 1,, τ ˆL m n) ( π y(t m 1 ) Y (t m 1 ) ) + + T (τ ˆL m 1,, τ ˆL m n, τ Ĝm (π y(t m 1 ))) T (τ ˆL m 1,, τ ˆL m n, τ Ĝm (Y m 1 )).

New efficient time integrators for non-linear parabolic problems 413 Tus, by applying (1) and (11), we obtain E m C q + C τ p+1 + ( e γ τ + C τ ) E m 1. From tis result is easy to conclude tat sceme (4) satisfies (5). 3. A NEW FOURTH ORDER LINEARLY IMPLICIT METHOD Now we present te main ideas of te construction process for a linearly implicit tree-level FSRK metod; suc metod attains fourt order in time wen it is applied to te numerical integration of te semilinear problems of type (1) combined wit spatial discretizations. To te best our knowledge, tis is te only fourt order metod of tis class wic appears in te literature. Te construction process for tis kind of metods is very complicated due to two main drawbacks; te first one is related to te ig number of order conditions tat we must impose, wic makes us cope wit large and complicated non-linear systems. Te second one is related to te additional requirements tat we must impose in order to guarantee te stability of te final metod. Precisely, te order conditions tat we must impose are (see [2]): (r i) A i e = C e for i = 1, 2, 3 (α 1) (b 1 ) T e = 1 (α 2) (b 1 ) T C e = 1 2 (α 3) (b 1 ) T A 1 C e = 1 6 (α 4) (b 1 ) T C C e = 1 3 (α 5) (b 1 ) T C C C e = 1 4 (α 6) (b 1 ) T C A 1 C e = 1 8 (α 7) (b 1 ) T A 1 C C e = 1 12 (α 8) (b 1 ) T A 1 A 1 C e = 1 24 (β 1) (b 2 ) T e = 1 (β 2) (b 2 ) T C e = 1 2 (β 3) (b 2 ) T A 2 C e = 1 6 (β 4) (b 2 ) T C C e = 1 3 (β 5) (b 2 ) T C C C e = 1 4 (β 6) (b 2 ) T C A 2 C e = 1 8 (β 7) (b 2 ) T A 2 C C e = 1 12 (β 8) (b 2 ) T A 2 A 2 C e = 1 24 (γ 1) (b 3 ) T e = 1 (γ 2) (b 3 ) T C e = 1 2 (γ 3) (b 3 ) T A 3 C e = 1 6 (γ 4) (b 3 ) T C C e = 1 3 (γ 5) (b 3 ) T C C C e = 1 4 (γ 6) (b 3 ) T C A 3 C e = 1 8 (γ 7) (b 3 ) T A 3 C C e = 1 12 (γ 8) (b 3 ) T A 3 A 3 C e = 1 24 ( ij) (b i ) T A j C e = 1 6 for {i, j} {1, 2, 3} ( ij) (b i ) T C A j C e = 1 8 for {i, j} {1, 2, 3} ( ij) (b i ) T A j C C e = 1 12 for {i, j} {1, 2, 3} ( ijk ) (b i ) T A 1j A k C e = 1 24 for {i, j, k} {1, 2, 3} One may ceck tat we need at least nine stages to obtain a linearly implicit metod wic fulfils tese order conditions. Restrictions (r 1 ), (r 2 ) and (r 3 ) require te assumption tat te first stage of te metod (C, A 1, b 1 ) is explicit; we must note tat in tis way te computational cost of te final metod will be similar to te cost of an eigt stage metod wit all stages implicit. To obtain a stable metod, it is convenient to impose te restrictions (,,, 1) T A i = (b i ) T for i = 1, 2, 3 (see [3]); in order to get te numerical solution directly from te calculation of te last stage Y m+1 (= Y m,9 ) and, consequently, to reduce ligtly te computational cost of te final metod, we impose te same restriction on te tird level (,,, 1) T A 3 = (b 3 ) T.

414 Blanca Bujanda, Juan Carlos Jorge Te computational cost of te final metod can be reduced (wen te coefficients of te linear operator do not depend on time) by assuming tat a 1 ii = a2 jj = a for i = 3, 5, 7, 9 and for j = 2, 4, 6, 8; besides, using tese restrictions we simplify te study of te stability of te FSRK metod. Tus, te table of te deduced metod will ave te following structure: C e A 1 A 2 (b 1 ) T (b 2 ) T = = c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 a 1 21 a a 1 31 a a 2 32 a 1 41 a 1 43 a 2 42 a a 1 51 a 1 53 a a 2 52 a 2 54 a 1 61 a 1 63 a 1 65 a 2 62 a 3 64 a a 1 71 a 1 73 a 1 75 a a 2 72 a 3 74 a 3 74 a 1 81 a 1 83 a 1 85 a 1 87 a 2 82 a 3 84 a 3 84 a b 1 1 b 1 3 b 1 5 b 1 7 a b 2 2 b 2 4 b 2 6 b 2 8 b 1 1 b 1 3 b 1 5 b 1 7 a b 2 2 b 2 4 b 2 6 b 2 8 A 3 (b 3 ) T = a 3 21 a 3 31 a 3 32 a 3 41 a 3 42 a 3 43 a 3 51 a 3 52 a 3 53 a 3 54 a 3 61 a 3 62 a 3 63 a 3 64 a 3 65 a 3 71 a 3 72 a 3 73 a 3 74 a 3 75 a 3 76 a 3 81 a 3 82 a 3 83 a 3 84 a 3 85 a 3 86 a 3 87 b 3 1 b 3 2 b 3 3 b 3 4 b 3 5 b 3 6 b 3 7 b 3 8 b 3 1 b 3 2 b 3 3 b 3 4 b 3 5 b 3 6 b 3 7 b 3 8 We begin te construction process for te metod by resolving te following order conditions: (r 1 ), (r 2 ), (α 1 ),, (α 8 ), (β 1 ),, (β 8 ), ( 12 ), ( 21 ), ( 21 ), ( 12 ), ( 112 ), ( 121 ) ( 211 ), ( 221 ), ( 212 ) and ( 122 ); wic involve te coefficients of te FSRK metods (C, A i, b i ) 2 only. Tus we obtain a four order family of FSRK metods depending on te free parameters a, c 4, c 5, c 6, c 7, a 1 65, a 1 83 and a 1 87. To perform, in an efficient way, te numerical integration of te linear stiff problems, it is necessary to impose additional restrictions of type A-stability. In order to introduce tis concept in te simplest way we apply a two-level FSRK metod to te numerical integration of te following test problem y (t) = (λ 1 + λ 2 ) y(t) wit Re(λ i ), for i = 1, 2;

New efficient time integrators for non-linear parabolic problems 415 obtaining te recurrence: y m+1 = ( 1 + 2 τ λ i (b i ) T ( I 2 τ λ j A j) 1 e )y m. By substituting in tis expression z i for τ λ i for i = 1, 2 we obtain a rational function of two complex variables; wic we call it te Amplification Function associated wit te FSRK metod 2 ( 2 R(z 1, z 2 ) = 1 + z i (b i ) T I z j A j) 1 e. We say tat an FSRK metod is A-stable iff R(z 1, z 2 ) 1, z i C being Re(z i ), i = 1, 2. As te FSRK metod as nine stages, wit te first one of tem explicit, and attains fourt order, te amplification function of te metod can be written as follows: R(z 1, z 2 ) = R 1 (z 1 )R 2 (z 2 ) + Rest were R i (z i ) = 1 + (1 4 a) z i + 1 2 (1 68 a + 12 a2 )z 2 i + ( 1 6 2 a + 6 a2 4 a 3 )z 3 i (1 a z i ) 4 + + ( 1 24 2a 3 + 3 a2 4 a 3 + a 4 )z 4 i (1 a z i ) 4 are te amplification functions associated wit te eac standard RK metod (C, A i, b i ) for i = 1, 2 and were: Rest = e 41 z 4 1 z 2 + e 32 z 3 1 z 2 2 + e 23 z 2 1 z 3 2 + e 14 z 1 z 4 2 + e 42 z 4 1 z 2 2 + e 33 z 3 1 z 3 2 (1 az 1 ) 4 (1 az 2 ) 4 + + e 24 z 2 1 z 4 2 + e 43 z 4 1 z 3 2 + e 34 z 3 1 z 4 2 + e 44 z 4 1 z 4 2 (1 az 1 ) 4 (1 az 2 ) 4, e 41 = E 4,1 1 24, e 14 = E 1,4 1 24, e 23 = E 2,3 1 12, e 32 = E 3,2 1 12, e 42 = E 4,2 4 a e 32 4 a e 41 1 48, e 24 = E 2,4 4 a e 23 4 a e 14 1 48, e 33 = E 3,3 4 a e 23 4 a e 32 1 36, e 43 = E 4,3 4 ae 42 1 a 2 e 41 4 a e 33 16 a 2 e 32 1 a 2 e 23 1 144, e 34 = E 3,4 4 ae 24 1 a 2 e 14 4 a e 33 16 a 2 e 23 1 a 2 e 32 1 144, e 44 = E 4,4 4 a e 43 4 a e 34 1 a 2 e 42 2 a 3 e 41 16 a 2 e 33 4 a 3 e 32 1 a 2 e 24 4 a 3 e 23 2 a 3 e 14 1 576,

416 Blanca Bujanda, Juan Carlos Jorge were E i1,i 2 = (b k1 ) T A k2 A kj e k (k 1,,k j) {1,2} j n l ( k)=i l, l=1,2 wit n l ( k) te number of te times tat te index l appears in te integer vector k. Given tis FSRK metod of Alternating Directions type attains fourt order, and te standard RK metod (C, A 1, b 1 ) may be reduced to a DIRK metod of five stages wit te first stage explicit, te standard RK metod (C, A 2, b 2 ) may be reduced to a four-stage SDIRK metod, we can see in [6] tat suc metods (C, A i, b i ) for i = 1, 2 are L-stable for a =.57281662482134746. Taking into account tat te strong stability and, particularly, te L-stability are very interesting properties wen we carry out te numerical integration of te parabolic problems wit irregular or incompatible data, we ave fixed a =.57281662482134746. Now, if we were able to make null te Rest we would obtain an L-stable FSRK metod. We ave cecked tat it is impossible to cancel te Rest because by substituting te previous values we observe tat e 44 as te constant value.11591. Neverteless, we can determine te remaining free parameters in order to minimize to some extent te contribution of te Rest to te amplification function. Due to te complexity, size and nonlinearity of te last expressions, we ave taken a 1 87 = in order to sorten tem; now, we can ceck tat e 14 = e 41, e 23 = e 32, e 24 = e 42 and e 43 = e 34, simplifying te subsequent study. By cancelling e 43 = e 34 and e 33 we can fix te parameters a 1 65 and a 1 83. By using tis option, te remaining coefficients of te Rest ave te following values: e 14 =.165332, e 23 =.171413 and e 24 =.423441. Wit te imposed restrictions, ave obtained a family of FSRK metods wic depend on te parameters c 4, c 5, c 6 c 7, attain fourt order and are A-stable. In te last part of te construction of tis metod we solve te remaining conditions. In suc order conditions, some coefficients of te last explicit level (C, A 3, b 3 ) appear. Namely we must solve: (C, A 3, b 3 ): (r 3 ), (γ 1 ),, (γ 8 ), ( 13 ), ( 23 ), ( 31 ), ( 32 ), ( 13 ), ( 23 ), ( 31 ), ( 32 ), ( 13 ), ( 23 ), ( 31 ), ( 32 ), ( 113 ), ( 131 ), ( 311 ), ( 331 ), ( 313 ), ( 133 ), ( 223 ), ( 232 ), ( 322 ), ( 332 ), ( 323 ), ( 233 ), ( 123 ), ( 132 ), ( 213 ), ( 231 ), ( 321 ) and ( 312 ). By resolving tese order conditions we obtain a family of metods wic depend on te parameters: a 3 63, a 3 73, a 3 74, a 3 76, a 3 82, a 3 83, a 3 84, a 3 85, a 3 86, a 3 87, c 4, c 5, c 6 and c 7. Suc coefficients are fixed in order to minimize to some extent te main term of te local error and also to simplify coefficients. Tus, te final metod is given by: A 1 (b 1 ) T = 1.572816624821349.572816624821349.572816624821349.233624731962216.1279261371115 =.14324156253764.14324156253764.572816624821349.247525445891478.114613915837628.5337551373812817.16584521264111465.16898133629871276.5743927888629.572816624821349 B C @.519764686433798.51822334268295.4843426395887455 A.657833985428.2245756812245128.45353475817523.6313327555445881.572816624821349.657833985421.2245756812245128.45353475817523.6313327555445881.572816624821349

New efficient time integrators for non-linear parabolic problems 417 A 2 (b 2 ) T = 1.572816624821349 1.1456321249642698.239482729148816.572816624821349.1122415459243.685561679411592.53637655875728.597488254772145.572816624821349.684549593831673.37263822447982.6584879946324 B C @ 1.189389864279955.77389697436742.33636848563513.572816624821349 A 1.23598239745338 1.73598239745343 1.73598239745338 1.23598239745343 1.23598239745338 1.73598239745343 1.73598239745338 1.23598239745343 A 3 (b 3 ) T = B @.572816624821349 1.14563212496414.364583822964946.2596334954886.165341413273 1.59735621929421 2.3353773662664 1.31892539518593.457158517861478.25963349528255.2363466549123.23975256771974143.4271839375178651 1.23598239745865 1.73598239745232 1.23598239745865 1.73598239745232.188919479934955.14276583987761113 1.7359823974529 1.23598239745358 1.7359823974529 1.23598239745358 1 C A Ce=(,.5 728 16 624 821 349,1.1 456 321 249 642 698, 1 3,.5 728 16 624 821 349, 2 3, 1 3,.427 183 937 517 865,1)T. 4. NUMERICAL EXPERIMENT In tis section we sow a numerical test were we ave integrated te following non-linear convection-diffusion problem wit a non-linear reaction term r(y) = k 1 y + k 2 y + y3 (1+y 2 ) 2 y t = y v 1 y x1 v 2 y x2 r(y) + f, (x 1, x 2, t) Ω [, 5], y(x 1,, t) = y(x 1, 1, t) =, x 1 [, 1] and t [, 5], y(, x 2, t) = y(1, x 2, t) =, x 2 [, 1] and t [, 5], y(x 1, x 2, ) = x 3 1(1 x 1 ) 3 x 3 2(1 x 2 ) 3, (x 1, x 2 ) Ω, wit Ω = (, 1) (, 1), v 1 = 1 + x 1 x 2 e t, v 2 = 1 + x 1 t, k 1 = k 2 = 1+(x1+x2)2 e t 2 and f = 1 3 e t x 1 (1 x 1 )x 2 (1 x 2 ). We ave carried out te time discretization using te metod described in te previous section and we ave discretized in space on a rectangular mes by using a central difference sceme. Te combination of metods of finite differences type of order q in te spatial discretization stage and pt order time integrators of te type described in tis paper provides a totally discrete sceme wose global error is [y(t m )] Y m = O(q + τ p ); ere [v] denotes te restriction of te function v to te mes node.

418 Blanca Bujanda, Juan Carlos Jorge In order to apply te integration metod described ere, we ave taken for i = 1, 2 and j = 1,, 9 L i (t m,j )Y m,j = δ xix i Y m,j [v i ] δˆxi Y m,j g (t m,j, Y m,j ) = 3(Y m)2 (Y m)4 (1 + (Y m)2 ) 3 Y m,j (Y m,j ) 3 (1 + (Y m,j ) 2 ) 2 [k i ] Y m,j 3(Y m )2 (Y m )4 2(1 + (Y m )2 ) 3 Y m,j, were δ xix i and δˆxi denote te classical central differences. Note tat, by using tis decomposition in eac stage of (4) we must solve only one linear system wose matrix is tridiagonal; tus te computational complexity of computing te internal stages of tis metod is of te same order as in te cases of an explicit metod. Note tat tese operators L i (t) are not commutative: neverteless, te numerical results obtained are correct. Te numerical maximum global errors ave been estimated as follows E N,τ = max Y N,τ (x 1i, x 2i, t m ) Y x 1i,x 2i,t m were Y N,τ (x 1i, x 2i, t m ) are te numerical solutions obtained at te mes point (x 1i, x 2i ) and te time point t m = m τ, on a rectangular mes wit (N + 1) (N + 1) nodes and wit time step τ, wile Y is te numerical solution calculated at te same mes points and time steps, but by using a spatial mes wit (2N + 1) (2N + 1) nodes, alving te mes size, and wit time step τ 2. We compute te numerical orders of convergence as log 2 E N,τ E 2N,τ/2. Table 1. Numerical Errors (E N,τ ) N = 8 N = 16 N = 32 N = 64 N = 128 N = 256 N = 512 2.2183E-2 5.692E-3 1.4231E-3 3.5569E-4 8.8921E-5 2.2289E-5 5.654E-6 In Table 1, we sow te numerical errors and, in Table 2, teir corresponding numerical orders of convergence. In order to guarantee tat te contribution to te error of te spatial and temporal part are of te same size, we ave assumed te relation Nτ C =.1 between te time step τ and te mes size 1 N. Table 2. Numerical orders of convergence N = 8 N = 16 N = 32 N = 64 N = 128 N = 256 1.9629 1.9995 2.3 2. 1.9962 1.9915

New efficient time integrators for non-linear parabolic problems 419 REFERENCES [1] B. Bujanda, J.C. Jorge, Stability results for linearly implicit Fractional Step discretizations of non-linear time dependent parabolic problems, to appears in Appl. Num. Mat. [2] B. Bujanda, J.C. Jorge, Order conditions for linearly implicit Fractional Step Runge Kutta metods (Preprint), Universidad Pública de Navarra. [3] B. Bujanda, J.C. Jorge, Efficient linearly implicit metods for nonlinear multidimensional parabolic problems, J. Comput. Appl. Mat. 164/165 (24), 159 174. [4] B. Bujanda, J.C. Jorge, Stability results for fractional step discretizations of time dependent coefficient evolutionary problems, Appl. Numer. Mat. 38, (21), 69 86. [5] G.J. Cooper, A. Sayfy, Aditive metods for te numerical solution of ordinary differential equations, Mat. of Comp. 35, (198), 1159 1172. [6] E. Hairer, G. Wanner, Solving ordinary differential equations II, Springer-Verlag, 1987. [7] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, 23. [8] N.N. Yanenko, Te metod of fractional steps, Springer, 1971. Blanca Bujanda blanca.bujanda@unavarra.es Universidad Pública de Navarra Dpto. Matemática e Informática Juan Carlos Jorge jcjorge@unavarra.es Universidad Pública de Navarra Dpto. Matemática e Informática Received: October 3, 25.