Some Applications of Fractional Step Runge-Kutta Methods

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1 Some Applications of Fractional Step Runge-Kutta Metods JORGE, J.C., PORTERO, L. Dpto. de Matematica e Informatica Universidad Publica de Navarra Campus Arrosadia, s/n 3006 Pamplona Navarra SPAIN Abstract: - In tis work we present some numerical metods wic belong to te class of Fractional Step Runge-Kutta scemes. Suc time integrators combined wit standard spatial discretization tecniques are capable of providing robust and very efficient algoritms for simulating many evolutive penomena wic are modelled by means of linear parabolic partial differential equations. Tese metods, wic can be considered as a generalization of classical Alternating Direction Implicit metods, can also be applied successfully wit oter approaces. We explain ere ow te same time integrators can be combined wit a suitable Domain Decomposition procedure obtaining numerical algoritms wic preserve te typical advantages of Domain Decomposition metods and, besides, avoid te use of Scwarz iterative processes. Two numerical experiments are sown in order to illustrate te teoretical qualities of tese scemes applied to te integration of two time dependent convection-diffusion-reaction problems. Key-Words: - Fractional Steps, Alternating Directions, Domain Decomposition, Additive Runge-Kutta, time dependent Convection-Diffusion Introduction In tis paper we tackle te efficient numerical resolution of multidimensional linear parabolic initial boundary value problems wose coefficients may depend on time. Suc problems use to be formuled as follows: find te function u : [t 0, T ] Ω R wic satisfies te equations du dt = A x, tu + f, x, t Ω t 0, T ], u x, t 0 = u 0 x, x Ω, Bu x, t = g x, t, x, t Ω t 0, T ], were Ω R n is te domain for te spatial variables and, for eac t t 0, T ], A x, t is an unbounded elliptic differential operator containing te derivatives of te unknown u wit respect to te spatial variables. We ave focused our experiences in several linear convection-diffusion reaction models were typically, A x, tu = d x, t u c x, t. u r x, tu, te diffusion term d is positive and te reaction term r is non negative. Operator B determines te type of boundary conditions considered Diriclet, Neumann,..., f is te source term, u 0 is te initial condition and g is te boundary data. We will assume enoug smootness and compatibility among all te data of problem in order to guarantee tat it possesses a sufficiently regular solution. It is well known tat a fine numerical integration of tese problems using classical tecniques is computationally expensive; if explicit metods are considered, due to stability restrictions, we will ave to restrict strongly te size of te time steps, specially wen fine meses are used to discretize te spatial variables; on te oter and, a classical implicit metod will work wit large time steps because it as no stability restrictions but its computational cost per time step will be very ig due to it will require te resolution of large linear sytems. Several alternative tecniques of integration ave been developed looking for preserving te unconditional stability property of classical implicit metods and reducing te computational cost of tem; one of te most celebrated advances in tis sense are te Alternating Direction Implicit ADI metods see [7], were only a set of uncoupled tridiagonal systems must be solved to advance in time. Wen complicated domains Ω are considered, and specially in a finite element context wic is te common option for te spatial discretization of tese problems, it is complicated to implement succesfully an ADI metod. In tis case a Domain Decomposition DD metod is a good option, specially if we dispose of parallel computers, because we can reduce a large linear system to a set of smaller linear systems wic can be solved in parallelsee [], altoug some additional cost must be assumed due to te iterative procedures p.e. Scwarz iterations wic are required to fit appropiately te boundary conditions in te subdomains. In tis paper we sow a generalization of te ADI metods and an improvement of te classical DD metods were no Scwarz iterations are required wic ave muc in common. Concretely, bot types of algoritms will result from te combination of a new type of time integrator called

2 Fractional Step Runge Kutta metod FSRK, joint to a suitable decomposition of elliptic operators A as a sum of simpler parts and a classical space discretization of tem via Finite Differences, Finite Elements,... Let us consider a decomposition for te elliptic operator and te source term in m addends as follows At = A i t, f = f i. 2 A time integration of problem using a FSRK metod provides us semidiscrete approximations to te exact solution in certain time moments u n x u x, t n by solving j U n,j = u n + τ a i k jk A ik x, t n,k U n,k k= +f ik x, t n,k, B ij U n,j = gx, t n,j, for j =,..., s, s u n+ = u n + τ b i j j A ij x, t n,j U n,j +f ij x, t n,j, 3 were i {,, m}, τ is te time step cosen for te integration, t n = nτ and t n,k = t n + c k τ k =,, s. We denote wit m te number of levels of te FSRK = number of addends in te decomposition and wit s te number of internal stages. Te advantages of tese metods wit respect to classical scemes come from te fact tat in 3 te operator acting implicitly in te j t internal stage is A ij x, t instead of te operator A x, t as it appens wen we use classical implicit scemes. So, a suitable coice of te splitting of operator A in simpler operators provides us scemes wic are more advantageous in some sense. If we follow te main idea of te ADI metods we will coose a splitting of A, in as many addends as te number of spatial variables taking part in our problem, in suc a way tat eac addend A i contains te derivatives of te unknown u wit respect to te spatial variable x i. Wen we complete a time semidiscretization of tis type wit a suitable discretization of te spatial variables, we come to an essentially one-dimensional system per internal stage instead of te multidimensional system obtained for eac stage in te case of using classical implicit metods. We ten acieve a reduction in te computational cost of te wole algoritm since te linear systems obtained wit tis tecnique involve tridiagonal matrices instead of te block tridiagonal or block pentadiagonal matrices arising from te use of classical implicit metods, depending on te dimension of te problem tat we are dealing wit is two or tree, respectively. It is clear tat a strong reduction of computational effort is obtained wit tis tecnique. In fact, a metod of tis class gets te same order of computational complexity per time step tat te corresponding to a classical explicit metod. Neverteless, as we pointed out previously, tere are some cases in wic tis tecnique is not te most appropriate one, for example if any crossed derivative appears in our equation, if te variation domain for te spatial variables is very complicated geometrically, or it is not going to be discretized by a rectangular mes. Anoter alternative, suitable for te previous cases, is to consider smoot splittings in 2 related to a decomposition of te spatial variables domain Ω see [8]. Let us consider te following domain decomposition Ω = m Ω i, were eac subdomain Ω i consists of a set of m i disjoint components Ω ij satisfying Ω i = m i Ω ij. In tis framework we consider A i x, t = ψ i xa x, t and f i x, t = ψ i xf x, t, were {ψ i x} m is a sufficiently smoot partition of unity. Eac function ψ i x takes te value 0 in points x wic do not belong to subdomain Ω i, takes te value in points x wic belong only to subdomain Ω i, and varies smootly between 0 and in te overlappings of subdomain Ω i wit te rest of subdomains. Wit tis kind of splitting te advantage wit respect to classical metods is not only based in te computational cost reduction of te algoritm but also in te possibility to parallelize some of te necessary calculations. Te reason for tis is tat we must solve in eac internal stage a linear system wic only possess unknowns in te mes points belonging to one of te domains Ω i and, due to te fact tat eac one of tese domains consists of a set of disjoint subdomains Ω ij, tis system can be seen as a collection of uncoupled linear subsystems one per eac disconnect component wic can be solved in parallel. It is interesting to notice tat applying te classical Domain Decomposition tecnique for parabolic problems see [] we obtain a similar algoritm but te difference between te two options is tat in te classical case we need to use an iterative Scwarz process wic is not necessary wen using te tecnique tat we are introducing ere. Te rest of te paper is divided in two sections: in te next one we give a brief revision to te main teoretical results wic ensure tat our scemes are unconditionally convergent and we write te totally discrete scemes in an abstract general formulation. Finally, in section 3 two numerical experiments are sown; te first one contains an appplication of a second order FSRK metod combined wit a splitting for te operator A of type ADI for a tree dimensional convection-diffusion problem, 2

3 wile in te second one te domain decomposition tecnique is used in combination wit te fractional implicit Euler metod to solve numerically a two dimensional diffusion-reaction problem. 2 Totally Discrete Scemes and Convergence Analysis: A Summary Te first key to perform an analysis of te convergence of a FSRK metod in a similar way to te classical Runge Kutta metods, is focused in te fact tat 3 can be formulated as an Additive RK sceme s U n,j = u n + τ a i jk A i x, t n,k U n,k k= +f i x, t n,k, B ij U n,j = gx, t n,j, for j =,..., s, s u n+ = u n + τ b i j A i x, t n,j U n,j +f i x, t n,j, were we ave extended te summatories considering many additional null coefficients; concretely, a i jk = 0 for k > j, ai jk = 0 for i i k and b i k = 0 for i i k. By grouping te coefficients of te metod in te following vectors and matrices b i = b i j Rs, C = diagc,, c s R s s, A i = a i jk Rs s, e =,, R s, we can express an Additive RK sceme of type in te compact form Ce A A 2... A m b T b T 2... b T m wic is analogous to a Butcer s table for a classical RK sceme. Wit tis notation it is easy to appreciate tat an Additive RK metod involves m standard RK scemes of certain order Ce A i b T i in suc a way tat eac one of tem determines te contribution of eac addend A i x, t and f i x, t of te splittings 2 to te numerical solution. 2. Order conditions In order to study te consistency of a FSRK sceme of type 3 we next introduce te concept of local error in time t n+ as ρ n+ = ut n+ ǔ n+,,, n = 0,,..., were ǔ n+ is te numerical solution obtained after one step of sceme 3 starting from te exact solution in te previous time moment ǔ n = ut n and we will say tat a FSRK metod is consistent of order p if, for sufficiently regular functions ut, it is satisfied tat ρ n+ C τ p+. In [] it is sown tat a FSRK sceme applied to solve a problem of type wit omogeneous boundary conditions, and suitable compatibility conditions among te source term end te initial and boundary conditions, is consistent of order p if te following order conditions are satisfied b T i C ρ A i2 C ρ 2... A ir C ρr e = r r j + +, r ρ k k=j i,..., i r {,..., m}, ρ,..., ρ r {0,..., p } suc tat r ρ k = i r, i {,..., p}, r {,..., i}. k= 5 In te case of considering time dependent boundary conditions, as we ave cosen in tis paper, a penomenon called order reduction appears. Tis beaviour was also observed in te cases of using a standard RK sceme see [2] or some classical Alternating Direction Implicit metods see [] as time integrators. In te case of using a FSRK sceme see [9] an order reduction from te classical order p to te order of te internal stages q occurs if te boundary conditions are time dependent, were q is te greater integer satisfying j A i C j e = C j e, i {,..., m}, j {,..., q}. 6 In suc paper a tecnique to avoid tis order reduction penomenon was also given; it as to be wit te modification of te classical boundary conditions considered for te internal stages. Te improved boundary conditions are defined as te evaluations of certain auxiliary functions Û n [j] in te corresponding boundaries, were suc functions are defined by means of te following recurrent process, for j Û [j] n Û [0] = e ut n + τ A i Ân i Û [j ] n + F n i, n = Ûn = ut n,,..., ut n,s T, 7 were Ân i = diaga i t n,,, A i t n,s, F n i = f i t n,,, f i t n,s T. In te same paper it was also proven tat te coice of te evaluations of 3

4 Û n [j+] in te corresponding boundaries permits us to recover one order of consistency wit respect to te one obtained wit Û n [j] for Û n [0] te order will be q until te classical order p is reaced. 2.2 Linear absolute stability Wit te aim of coming to a convergence result for te semidiscrete sceme 3, we need to complete te previous result on consistency wit a suitable stability property. Te procedure to follow now is analogous to te case of considering a standard RK sceme; if we apply our sceme to te test scalar initial value problem m u t = λ i ut, ut 0 = u 0, we obtain te recurrence u n+ = τλ i b T i I + τλ j A j e u n. Substituting τλ i by z i we define te amplification function associated to 3 as te following rational function depending of m complex variables z, z 2,, z m Rz,, z m = z i b T i I + z j A j e and we say tat our FSRK metod is A-stable iff its associated amplification function satisfies R z, z {z,, z m : z i C and Rez i 0, i =,, m}. In [2] it is proven tat, under suitable properties for te operators A i x, t, te use of an A-stable FSRK sceme guarantees a stable beaviour for te time discretization process. 2.3 Te totally discrete scemes: Convergence Finally, to get a numerical algoritm we need to discretize te stationary problems arising in 3 wit respect to te spatial variables in a suitable way. In [3], [9] te autors propose an abstract formulation for a general discretization metod and te consistency and stability properties wic it must satisfy to obtain te appropiate convergence result for our scemes. Tese properties are satisfied for most of standard discretization metods of elliptic problems Finite Differences, Finite Elements, Spectral Metods,... Using any of tese metods to discretize in space we obtain a totally discrete sceme wic as te following form: U n,j = u n, + τ j k= a i k jk A ik t n,k U n,k +f ik t n,k, B ij U n,j = g t n,j, for j =,..., s, s u n+, = u n, + τ b i j j A ij t n,j U n,j +f ij t n,j, 8 In tis formulation is te mes discretization parameter and u n+, denotes te numerical approximations to u x, t n+ wic we are going to compute and wic are defined of finite dimensional vector spaces wose dimensions grow to infinity wen tends to zero. Typically, te vectors u n+, will contain values in te nodes of a mes in a Finite Difference context, or a continouous piecewise polynomial function in a Finite Element context. For eac value of we ave tat te operators A i And similarly for B i are going to be approximated by linear operators A i, in a consistent and stable form. Tis means tat A i, tr tv π A i tv = O r, 9 for functions v x wic are sufficiently smoot in Ω applications r, π are restricitons to te mes nodes in Finite Differences and appropiate projections to spaces of continuous piecewise polynomial functions in a Finite element framework, and also tat { I c Ai, v = w, c > 0 0 B i, v = w b,, ave unique solution continuously dependent of te data w, w b, uniformly in. Finally, combining te last two properties, joint to te consistency an absolute stability of te FSRK sceme, te following unconditional convergence result can be obtained see [9] r t n ut n u [j],n = O r + τ p and we will say in tis case tat te metod is unconditionally convergent of order r in space and of order p in time, 3 Numerical Experiments 3. A 3D Application of type ADI We want to solve a problem of type posed in te domain Ω [t 0, T ] = 0, 3 [0, 2] wit Diriclet boundary conditions; te cosen diffusion coefficient considered is d x, t = + e t xyz, te

5 velocity field is c x, t =,, and te reaction coefficient r x, t is null. Data f, u 0 and g are calculated in order to tis problem admits ux, y, z, t = e +xyz t as exact solution. For its numerical resolution we combine te second order FSRK sceme proposed in [0] as time integrator wit a classical central difference sceme for te spatial discretization. Following te main ideas of Alernating Direction Metods, te splitting 2 cosen for te elliptic operators A x, t is te following one: A x, t = d x, t 2 x 2 x A 2 x, t = d x, t 2 y 2 y A 3 x, t = d x, t 2 z 2 z f i x, t = u 3 t A i x, tu, i =, 2, 3. Te global errors in te maximum norm E [j],τ = max ut n, x i, y k, z l u [j] n,i,k,l,n are displayed in Table Nτ = 0.8, wit N =. We include in te first row te global error obtained in te case of considering classical boundary conditions for te internal stages u [0],n u,n; in tis case we find an order reduction to te order of te internal stages wic is 0 for tis sceme. In te second row we write te global errors for u [],n wic is obtained wit te same numerical algoritm, but canging te classical boundary conditions by evaluations of Û n [] in te nodes of te boundary. In tis case only a reduction of order in one unity is observed and in te last row we write te blobal errors obtained in te case of coosing evaluations of Û n [2] in te nodes of te boundary, avoiding completely te order reduction. From tis global errors te numerical orders of convergence are calculated as usual O [j] log 2 E [j],τ E [j],τ/2,τ = and te obtained results are also sown, below te global errors, in Table. 3.2 A 2D application of type Domain Decomposition We consider now te following two-dimensional problem du dt = u + sinπx sinπyt u, ux, y, 0 = sinπx sinπy, x, y Ω, ux, y, t = 0, x, y, t Ω 0, 2], posed in te domain Ω [t 0, T ] = 0, 2 [0, 2], for wic we do not know te analytical expression of its exact solution. N=8 N=6 N=32 N=6 N=28 j= E E E E E E-.838E E E-5.02E j=2.730e- 5.92E-5.73E E-6.37E Table : Global errors and convergence orders of example In tis example we consider a splitting for te elliptic operator wic is associated to a decomposition of te unit square in four subdomains Ω i, i =, 2, 3,, eac one of tem consisting of disjoint components Ω ij, j =, 2, 3, see Figure. Ω 33 Ω 3 Ω 3 Ω Ω 3 Ω 23 Ω Ω 2 Ω 3 Ω Ω 32 Ω 2 Ω Ω 2 Ω 2 Ω 22 Figure : Domain decomposition For doing tat, we introduce te following functions of one variable, if x [0, d] [ 2 + d, 3 d], 0, if x [ + d, 2 d] [ 3 + d, ], i x = 2 3 d x α + x α 3, d 3 if x [α d, α + d], α =, 2, 3 and i 2 x = i x and we define te following partition of unity in Ω: ψ x, y = i xi y, ψ 2 x, y = i 2 xi y, ψ 3 x, y = i xi 2 y and ψ x, y = i 2 xi 2 y, were te cosen size for te overlapping zones as been d = 8. Using tis partition we ave done te splitting for te operator A and te source term f A i = ψ i A, f i = ψ i f, i =, 2, 3, as we previously suggested. In tis case te numerical algoritm as been obtained by combining again a clas- 5

6 sical central difference sceme wit te Fractional Implicit Euler metod u n+ i = u n+ i + τa i x,t n+ u n+ i +f i x,t n+, i =, 2, 3,, n = 0,,... 2 wic is convergent of fist order. As we do not know te exact solution of te continuous problem, in tis case we ave estimated te global errors using te double mes principle max n,i,j uτ n, i, j u τ 2 i, j, were u τ n, n, 2 is te numerical solution obtained using our sceme E [j],τ wit time step τ and mes size and u τ 2 is te numerical solution obtained after two steps of size τ 2 in n, 2 a mes wose size is 2 ; we compare bot numerical solutions in te points of te coarser grid of size. Te global errors obtained are sown in Table 2. As te totally discrete sceme is convergent of order 2 in space and order in time we ave cosen te time step for different mes sizes = /N satisfying te relation N 2 τ = CC = 5.2, wit te aim of preserving contributions to te global error of te same order for bot te spatial and te time discretization stages. We also include in tis table te estimated numerical orders of convergence. N=6 N=32 N=6 N=28 N= E E E E E Table 2: Global errors and convergence orders of example 2 References: [] B. Bujanda, Métodos Runge-Kutta de Pasos Fraccionarios de orden alto para la resolución de problemas evolutivos de conveccióndifusión-reacción, Tesis, Universidad Pública de Navarra, 999. [] G. Fairweater, A.R. Mitcell, A new computational procedure for A.D.I. metods, SIAM J. Numer. Anal., Vol., 967, pp [5] J.C. Jorge, B. Bujanda, Tird order fractional step metods for multidimensional evolutionary convection-diffusion problems. Finite difference metods: teory and applications Rousse, 997, Nova Sci. Publ., Commack, NY, 999, pp [6] J.C. Jorge, B. Bujanda, Hig Order Fractional Step Metods For Evolutionary Convection Diffusion Reaction Problems. Problems in Modern Applied Matematics, 2000, World Sci. and Eng. Soc Press, pp [7] G.I. Marcuk, Splitting and Alternating Direction Metods. Handbook of Numerical Analysis Vol., Edited by P.G. Ciarlet and J.L. Lions, Nort Holland, 990, pp [8] T.P. Matew, P.L. Polyakov, G. Russo, J. Wang, Domain Decomposition Operator Splittings for te Solution of Parabolic Equations. SIAM J. Sci. Comput., Vol. 9, No. 3, 998, pp [9] L.Portero, J.C. Jorge, B. Bujanda, Avoiding order reduction of Fractional Step Runge-Kutta metods for linear time dependent coefficient parabolic problems, Appl. Numer. Mat., Vol. 8, 200, pp [0] L. Portero, J. C. Jorge, A Generalization of Peaceman & Racford Fractional Step Metod, Preprint Universidad Pública de Navarra, Dpto. Matemática e Informática, Spain, 200. [] A. Quarteroni, A. Valli, Domain Decomposition Metods for Partial Differential Equations, Clarendon Press, Oxford 999 [2] J.M. Sanz-Serna, J.G. Verwer, W.H. Hundsdorfer, Convergence and order reduction of Runge- Kutta scemes applied to evolutionary problems in partial differential equations, Numer. Mat., Vol. 50, 986, pp [2] B. Bujanda, J.C. Jorge, Stability results for fractional step discretizations of time dependent coefficient evolutionary problem, Appl. Numer. Mat., Vol. 38, 200, pp [3] B. Bujanda, J.C. Jorge, Fractional Step Runge- Kutta metods for time dependent coefficient parabolic problems, Appl. Numer. Mat., Vol. 5, 200, pp