ON THE PERFORMANCE OF LOW

Transcription

1 Monografías Matemáticas García de Galdeano, (6) ON THE PERFORMANCE OF LOW STORAGE ADDITIVE RUNGE-KUTTA METHODS Inmaclada Higeras and Teo Roldán Abstract. Gien a differential system that inoles terms with different stiffness properties, a natral approach to obtain nmerical approximations is the se of implicit-explicit time-discretizations. These systems, often with a large nmber of eqations, arise from the semidiscretization of some time-dependent partial differential eqations. In the constrction of Rnge-Ktta methods, properties like stability and accracy are important items that mst be taken into accont. Howeer, in some contexts, storage reqirements of the schemes also play an important role. When the high dimension of the problem compromises the compter memory capacity, it is important to incorporate low memory sage to other properties of the scheme. In a recent work the athors hae stdied and constrcted implicit-explicit Rnge- Ktta methods with low-storage reqirements. In this paper we deelop and test new low-storage Rnge-Ktta methods that complete the stdy done in the mentioned work. Keywords: Additie Rnge-Ktta methods, low-storage, stiff problems. AMS classification: 65L, 65L6, 6L.. Introdction Space discretization of some time-dependent partial differential eqations (PDEs) gies rise to systems of ordinary differential eqations in additie form. These systems, often with a large nmber of eqations, arise, e.g., from semidiscretisations of conection-diffsion problems and hyperbolic systems with relaxation [, 5,, 5, 6]. When the differential system inoles terms with different stiffness properties, a natral approach to obtain nmerical approximations is the se of implicit-explicit (IMEX) timediscretizations. IMEX Rnge-Ktta methods hae been deeply stdied in the literatre (see, e.g., [,,,, 5, 9]). Sometimes, the additie differential system is of the form = f (, ), = f (, ) + ε g (, ), () where ε is the stiffness parameter. These systems hae been considered, e.g., in [,,,,, 5, 6], where robst IMEX Rnge-Ktta methods hae been analyzed. In particlar, in [, ], niform conergence in the stiffness parameter ε has been stdied. For systems with a large nmber of eqations, memory storage reqirement is an important isse. When the high dimension of the problem compromises the compter memory

2 78 Inmaclada Higeras and Teo Roldán capacity, it is important to incorporate low memory sage to some other properties of the scheme. These ideas hae been deeloped in [6, 7, 8, 9,,,,, 7, 8], where different low-storage Rnge-Ktta methods hae been constrcted. In particlar, in [6, ], explicit Rnge-Ktta methods are implemented by sing the an der Howen format [7], while diagonally implicit Rnge-Ktta (DIRK) schemes hae been explored in []. Robst IMEX Rnge-Ktta methods with low-storage reqirements were recently constrcted in []. In particlar, new Additie Semi-Implicit Rnge-Ktta methods (ASIRK), a special class of IMEX Rnge-Ktta methods, hae been deeloped. These new methods can be implemented by sing jst three memory registers. In this paper we complete that work constrcting new low-storage methods. The rest of the paper is organized as follows. In section we briefly introdce ASIRK methods, a special class of IMEX Rnge-Ktta schemes. In section we reiew low-storage ASIRK implementations in three memory registers. Taking into accont the reslts obtained in [], we constrct new low-storage ASIRK schemes in section. Some nmerical experiments are displayed in section 5.. Additie Semi-Implicit Rnge-Ktta methods We consider a special class of IMEX Rnge-Ktta schemes that hae been sed in [9] looking for comptational efficiency. These schemes are sally referred as ASIRK-sA methods. The nmerical soltion of a general additie differential problem with an ASIRK-sA method is gien by y = f (y) + g(y), y(t ) = y, () y n+ = y n + s ω i K i,n+, () i= where the internal deriaties K i,n+ are gien by ( K i,n+ = h f (y n + i i b i j K j,n+ ) + g(y n + j= j= ) c i j K j,n+ + c ii K i,n+ ), i =,..., s, () and b i j, c i j, ω j are the coefficients of the method. Obsere that the reslting scheme is explicit in f and diagonally implicit in g. On the following, we denote the matrices and ector containing the coefficients of the ASIRK-sA method by B = (b i j ), C = (c i j ), and ω, respectiely. Below we display these matrices for s =, c ω B = b, C = c c, ω = ω. (5) b b c c c ω Een thogh method ()-() can be considered as an additie Rnge-Ktta method, it is important to notice that the coefficients in (5) are not the standard coefficients of an additie Rnge-Ktta scheme. Howeer, as we show below, eqations ()-() can be rewritten as an

3 On the performance of low storage Additie Rnge-Ktta methods 79 additie Rnge-Ktta method (9). With this approach, some isses for ASIRK-sA methods, e.g., the set of order conditions, can be obtained in a simple way from the theory of additie Rnge-Ktta methods. Eqations ()-() are gien in terms of K n+ = (K t,n+,..., Kt s,n+ )t, the internal deriatie ector. By sing the Kronecker prodct and denoting K n+ = h ( F(Y n+ ) + G(Ŷ n+ ) ), where F(Y n+ ) = ( f (Y,n+ ) t,..., f (Y s,n+ ) t ) t and G(Ŷ n+ ) = (g(ŷ,n+ ) t,..., g(ŷ s,n+ ) t ) t, it is possible to rewrite ()-() as y n+ = y n + h (ω t I k ) ( F(Y n+ ) + G(Ŷ n+ ) ), (6) where the internal stages Y n+ = (Y t,n+,..., Yt s,n+ )t and Ŷ n+ = (Ŷ t,n+,..., Ŷt s,n+ )t are obtained from Y n+ = e y n + (B I k ) ( h F(Y n+ ) + h G(Ŷ n+ ) ), (7) Ŷ n+ = e y n + (C I k ) ( h F(Y n+ ) + h G(Ŷ n+ ) ). (8) This formlation corresponds to a s-stage additie Rnge-Ktta method, where the s internal stages are Y n+ and Ŷ n+. Then, the doble Btcher tablea of scheme (6)-(8) is Be B Ce C ω t B C (9) ω t Now, from the set of order conditions for IMEX methods, it is easy to derie the set of order conditions for ASIRK-sA scheme ()-() in terms of matrices B, C, and ector ω (see [] for details). In the same way, it is posible to derie stability properties of ASIRK-sA methods ()-() applying some known reslts on linear stability for additie Rnge-Ktta methods [, 5, 9]; or approach is similar to the one made in [5]. Ths, we consider the simplified linear model eqation y = ξ y + ξ y, y() =, () where ξ, ξ C, with Re (ξ ) <, and we sole the model eqation () with method ()-(). In this way, we can write the nmerical soltion as y = R(z, z ), where the fnction of absolte stability R(z, z ) is obtained in terms of the coefficients of the method and z = ξ h, z = ξ h. We are interested in obtaining the largest set S = {z C : sp z C R(z, z ) }. () Besides, in order to get a correct asymptotic decay for the stiff terms, the L-stability condition for the implicit step, that is, lim R(, z ) =, () z

4 8 Inmaclada Higeras and Teo Roldán mst be imposed. For more details, see [5]. We are also interested in stiff accracy, that is, accracy of the nmerical soltion of problem () for small ales of ε. This isse has been stdied in [, 5] for IMEX Rnge- Ktta methods. The analysis done leads to the following additional order conditions for an IMEX Rnge-Ktta method with coefficients (A, b t, c), (Ã, b t, c), b t A c =, b t A c =, b t A Ã c = /. () For ASIRK-sA methods, see [] for details, we get these additional orden conditions ω t e =, ω t C (Ce) =, ω t B e = /. () The linear system = δ + σ, = δ + σ + ε (c ), (5) is a particlar case of (). If we consider consistent initial ales, = c, then, after one time step, the exact soltion of (5) satisfies (ε, h) = ( + â h + â h + σ ˆb h ε ) + O(h, h ε ), (ε, h) = ( ( + â h + â h ) c + ˆb ε + (â + σ c) ˆb h ε ) + O(h, h ε ), where, as in [5], we hae denoted â = δ + σ c, and ˆb = δ + (σ δ ) c σ c. We will constrct the nmerical soltion ( (ε, h), (ε, h)) of system (5) with the ASIRK-sA scheme ()-(), and from the difference between the nmerical and the exact soltion (ε, h) (ε, h), (ε, h) (ε, h), (6) we will get conditions on the coefficients of the method.. ASIRK methods implemented in three memory registers In this section we consider a kind of ASIRK-sA methods that can be implemented in an efficient way by sing three memory registers for any nmber of stages. The way this schemes hae been deeloped can be seen with details in []. The coefficients of these methods hae a particlar strctre that is mainly enforced by the possibility of implementing the scheme in three memory registers. The ASIRK-sA methods we propose are of the form λ ω ω +γ ω λ ω B = ω ω +γ, C = ω ω λ, ω =. (7) ω ω... ω s +γ s ω ω... ω s λ s ω s

5 On the performance of low storage Additie Rnge-Ktta methods 8 For example, a low-storage ASIRK-A method can be displayed in this way K = h f (y n ) + h g(y n + λ K ), K = h f (y n + ω K + γ K ) + h g(y n + ω K + λ K ), K = h f (y n + ω K + ω K + γ K ) + h g(y n + ω K + ω K + λ K ), y n+ = y n + ω K + ω K + ω K. For each stage, a memory register (Register ) is sed for the storage of y n + i j= ω j K j, Register : Y i = Y i + ω i K i, i =,..., s +, where we consider b = and Y = y n. After the last stage, we obtain the nmerical soltion y n+ in the Register as Y s+ = Y s + ω s K s. The second memory register (Register ) is sed for the storage of the ealation of the fnction f, Register : L i = h f (Y i + γ i K i ), i =,..., s, where γ =. Finally, the third memory register (Register ) is sed for the internal deriatie K i Register : K i = L i + h g(y i + λ i K i ), i =,..., s. (8). A new method implemented in three memory registers Gien a low-storage ASIRK-sA method of the form (7), we determine its coefficients by imposing accracy, stiff accracy and stability properties. In [] the cases of two stages, s =, and three stages, s =, were stdied in detail. In this paper we consider s = and we make a different choice of one of the free parameters. Obsere that if λ s = ω s in (7), then the implicit scheme is stiffly accrate. Besides, in this case, the implicit part of the method satisfies condition (). Accordingly, we set λ = ω and we impose the first order condition ω t e =, that is, ω = ω ω. With these choices, from schemes (7) we get a family of ASIRK-A methods with 6 parameters, λ ω B = ω + γ, C = ω λ, ω = ω. (9) ω ω + γ ω ω ω ω ω ω For methods (9), second order is obtained if the parameters satisfy ω (ω + γ ) + ( ω ω + )(ω + ω + γ ) =, ω ω + ω λ ω + ω λ ω + =. (a) (b) It can be checked that, with the 6 parameters in (9), it is not possible to achiee order three. For second order ASIRK-A methods of the form (9), the additional order conditions () are satisfied; obsere that in (9), the last row of C is eqal to ω.

6 8 Inmaclada Higeras and Teo Roldán Frthermore, in (9) we set λ = λ = λ. In this way, if fnction g is linear, or if we se Newton-like methods for soling the nonlinear systems, the same LU-factorization can be sed when the first two internal deriaties are compted (see eqation (8)). From the stiff accracy analysis (6), cancellation of the term εh in the ariable, and the term ε /h in the ariable, leads to the eqations ω λ(ω + γ ) + (ω + ω ) (ω (ω + γ ) λ(ω + ω + γ )) =, λ () (ω λ)(ω λ) λ (ω + ω ) =. () If () and () hold, the difference between the nmerical and the exact soltion of problem (5) is of the form (ε, h) (ε, h) = O(h, h ε), (ε, h) (ε, h) = O(h, h ε). () In order to impose condition (), there exists two posibilities: λ = ω or λ = ω. In [] only the case λ = ω was stdied. If we sbstitte this ale in the other three eqations, namely, (a), (b) and (), and we sole the corresponding system, we get a ω -family of second order methods sch that the implicit method is L-stable, proided that it is A-stable. Frthermore, the methods satisfy the additional conditions (), they can be implemented by sing jst three memory registers and, for the model problem (5), the errors are gien by (). In order to choose the parameter ω in this family of methods, different approaches can be followed. If we minimized the local trncation error, then we obtain the scheme named ASIRK-LSe(,). For more details see []. In this paper we stdy the other posibility, that is λ = ω in (). If we sbstitte this ale in the other three eqations (a), (b) and (), then, from the corresponding system we get γ = ω5 ω + ω 6ω + ω ω ω + ω 6ω +, ω = ω + ω (ω ), () γ = 8ω6 7ω5 + ω ω + 6ω ω + ω (ω ) ( ω ω + ). This is another ω -family of second order methods with the same properties as the family obtained for the case λ = ω in []. We can follow again different approaches in order to choose the parameter ω in (). We can minimize the local trncation error, bt also we can optimize the stability region (). In both cases the deried methods hae a similar behaior, so in this work we will only show the method obtained by minimizing the local trncation error.

7 On the performance of low storage Additie Rnge-Ktta methods 8 ASIRK-LSe(,) method We hae denoted by ASIRK-LSe(,) the method in () that minimizes the sm of the absolte ales of the coefficients in the leading trncation error term. The optimm ale for this sm is.56, and it is obtained for ω =.88. We remark that the optimm ale is significantly higher than the one obtained for the method ASIRK-LSe(,) in [], namely.8. A rational approximation of ω =.88 is ω = /7. In that case we obtain the method. B = , C = Nmerical Experiments, w = (5) In this section we stdy the performance of the nmerical scheme constrcted in this paper, named ASIRK-LSe(,), and we compare it with the method ASIRK-LSe(,) gien in [], and the ASIRK-A method considered by Zhong in [9]. In [] we tested the methods on different problems bt here we will jst see a simple prototype of stiff system of the form =, = + ε (e() ), (6) with e() = sin. We can choose arbitrarily the initial ale (), bt there is no freedom in the choice of (). If we consider non-consistent initial ales, then the soltion presents an initial layer in the component when ε. We hae chosen () = π/ and we hae integrated the problem in [, ], with time step h =.5, assming different initial data (). Consistent initial ales are obtained if we take () = sin(()) =. By adding a small pertrbation δ, we get non-consistent initial ales; in the nmerical experiments we hae taken δ =.5. Finally we hae also considered well prepared initial data to obtain a smooth soltion. C InVal : () =, NC InVal : () = + δ, (7) WP InVal : () = + π ε π ε. For each method we compte the conergence rates for a wide range of ales of the parameter ε; in or tests we consider ε = j, j =,,,..., 6. We try to check whether the conergence is niform in ε, particlarly in the intermediate regime. In order to show the conergence rates, for each ale of ε we compte E h and E h/, an estimation of the relatie L -global error with stepsizes h and h/, respectiely. With these ales we compte the conergence rate in the standard way.

8 8 Inmaclada Higeras and Teo Roldán ASIRK LSe(,) ASIRK LSe(,) ASIRK LSe(,) 5 ASIRK LSe(,) 5 ASIRK LSe(,) 5 ASIRK LSe(,) 5 Zhong s ASIRK A 5 Zhong s ASIRK A 5 Zhong s ASIRK A Figre : Conergence rates of (continos line) and (dashed line) s. stiffness parameter ε for problem (6) for stepsize h =.5. From top to bottom: ASIRK-LSe(,) (5), ASIRK-LSe(,) and Zhong s method. From left to right: NC InVal, C InVal and WP InVal initial ales (7). In figre we show the conergence rates erss the stiff parameter ε for the system (6), when methods ASIRK-LSe(,), ASIRK-LSe(,) and Zhong s method are sed. For C InVal and NC InVal the new method ASIRK-LSe(,) gies reslts similar to the ones obtained with ASIRK-LSe(,) scheme. Howeer, for WP InVal initial conditions, the new method in this paper has a worse behaior in the middle stiff regime, that is, when h is close to ε. These poor reslts may be de to the higher ale of the leading trncation error terms. Conseqently, the choice of λ = ω does not lead to a nmerical scheme better than the ones obtained in []. Acknowledgements Spported by Ministerio de Economía y Competiidad, project MTM-.

9 On the performance of low storage Additie Rnge-Ktta methods 85 References [] Ascher, U., Rth, S.,, and Spiteri, R. Implicit-explicit Rnge-Ktta methods for time-dependent partial differential eqations. Appl. Nmer. Math. 5 (997), [] Boscarino, S.,, and Rsso, G. On a class of niformly accrate IMEX Rnge-Ktta schemes and applications to hyperbolic systems with relaxation. SIAM Jornal on Scientific Compting (9), [] Boscarino, S. Error analysis of IMEX Rnge-Ktta methods deried from differentialalgebraic systems. SIAM Jornal on Nmerical Analysis 5 (7), 6 6. [] Boscarino, S. On an accrate third order implicit-explicit Rnge-Ktta method for stiff problems. Applied Nmerical Mathematics 59 (9), [5] Caflisch, R. E., Jin, S., and Rsso, G. Uniformly accrate schemes for hyperbolic systems with relaxation. SIAM Jornal on Nmerical Analysis (997), 6 8. [6] Calo, M., Franco, J.,, and Rández, L. Minimm storage Rnge-Ktta schemes for comptational acostics. Compters & Mathematics with Applications 5 (), [7] Carpenter, M. H., and Kennedy, C. A. Forth-order n-storage Rnge-Ktta schemes. NASA Technical Memorandm 9 (99). [8] Gottlieb, S., and Sh, C. Total ariation diminishing Rnge-Ktta schemes. Math. Comp. 67 (998), [9] Happenhofer, N., Koch, O., and Kpka, F. IMEX Methods for the ANTARES Code. ASC Report 7 (). [] Higeras, I., and Roldán, T. Constrction of robst implicit-explicit Rnge-Ktta methods with low-storage reqirements. Sbmitted to Jornal of Scientific Compting (). [] Kennedy, C. A., and Carpenter, M. H. Additie rnge-ktta schemes for conectiondiffsion-reaction eqations. Applied nmerical mathematics (), 9 8. [] Kennedy, C. A., Carpenter, M. H., and Lewis, R. M. Low-storage, explicit Rnge-Ktta schemes for the compressible Naier-Stokes eqations. Applied nmerical mathematics 5 (), [] Ketcheson, D. I. Highly efficient strong stability-presering Rnge-Ktta methods with low-storage implementations. SIAM Jornal on Scientific Compting (8), 6. [] Koch, O., Kpka, F., Löw-Baselli, B., Mayrhofer, A., and Zassinger, F. SDIRK Methods for the ANTARES Code. ASC Report (). [5] Pareschi, L., and Rsso, G. Implicit-explicit Rnge-Ktta schemes for stiff systems of differential eqations. Recent Trends in Nmerical Analysis (), [6] Pareschi, L., and Rsso, G. High order asymptotically strong-stability-presering methods for hyperbolic systems with stiff relaxation. Hyperbolic Problems: Theory, Nmerics, Applications (), 5.

10 86 Inmaclada Higeras and Teo Roldán [7] Van Der Howen, P. J. Constrction of integration formlas for initial ale problems. North Holland, 977. [8] Williamson, J. Low-storage Rnge-Ktta Schemes. Jornal of Comptational Physics 5 (98), [9] Zhong, X. Additie semi-implicit Rnge-Ktta Methods for compting high-speed noneqilibrim reactie flows. Jornal of Comptational Physics 8 (996), 9. Inmaclada Higeras and Teo Roldán Dpto. Ing. Matemática e Informática Uniersidad Pública de Naarra higeras@naarra.es and teo@naarra.es