×

Partitioned exponential methods for coupled multiphysics systems. (English) Zbl 1462.65130

New partitioned exponential integrators are developed for coupled multiphysics problems by changing the information between components of the coupling terms. Two different approaches are considered for the construction and analysis of these methods, one based on splitting the component functions into linear and nonlinear terms (split-RHS methods), and the other is based on approximating the Jacobians of individual components (W-type methods). Two new formulations of partitioned exponential methods are proposed: partitioned methods of Rosenbrock-exponential type (PEXPW), and partitioned Runge-Kutta style exponential methods (PEPIRKW).
The matrix-exponential-like functions of the proposed partitioned methods are evaluated on the Jacobians of the individual component functions, whereas these functions are applied to the full (coupled) Jacobian in an unpartitioned method. If the individual Jacobians have computationally favorable structures, then the computational expense of evaluating matrix-exponential-like functions is greatly reduced. In reaction-diffusion systems with two or more species, the diffusion Jacobian is block-diagonal, the matrix-exponential-like functions can be evaluated on individual blocks in parallel. The numerical experiments show that using this strategy the partitioned exponential methods are at least twice as fast as the unpartitioned counterpart. One drawback of the partitioned exponential methods is the reduced stability. The numerical tests illustrate that the partitioned methods can exhibit more stable behavior than an unpartitioned method in some stiffness regimes, wheras in some very stiff regimes, partitioned methods can fail to obtain a solution.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L04 Numerical methods for stiff equations
15A16 Matrix exponential and similar functions of matrices
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akrivis, Georgios; Crouzeix, Michel, Linearly implicit methods for nonlinear parabolic equations, Math. Comput., 73, 246, 613-635 (2004) · Zbl 1045.65079
[2] Allen, Samuel M.; Cahn, John W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27, 6, 1085-1095 (1979)
[3] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T.R., Implicit explicit methods for time-dependent partial-differential equations, SIAM J. Numer. Anal., 32, 797-823 (1995) · Zbl 0841.65081
[4] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 151-167 (1997) · Zbl 0896.65061
[5] Augustine, A.; Sandu, A., MATLODE: a Matlab ODE solver and sensitivity analysis toolbox
[6] Belytschko, T.; Yen, H.-J.; Mullen, R., Mixed methods for time integration, Comput. Methods Appl. Mech. Eng., 17-18, 259-275 (Feb 1979) · Zbl 0403.73002
[7] Bhatt, Ashish; Moore, Brian E., Structure-preserving exponential Runge-Kutta methods, SIAM J. Sci. Comput., 39, 2, A593-A612 (2017) · Zbl 1365.65271
[8] Butcher, J. C., An algebraic theory of integration methods, Math. Comput., 26, 117 (Jan 1972), 79-79 · Zbl 0258.65070
[9] Butcher, J. C., Trees and numerical methods for ordinary differential equations, Numer. Algorithms, 53, 2-3, 153-170 (Mar 2009) · Zbl 1184.65072
[10] Butcher, J. C., Trees, B-series and exponential integrators, IMA J. Numer. Anal., 30, 131-140 (2010) · Zbl 1185.65121
[11] Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2016), Wiley · Zbl 1354.65004
[12] Calvo, M. P.; de Frutos, J.; Novo, J., Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations, Appl. Numer. Math., 37, 4, 535-549 (2001) · Zbl 0983.65106
[13] Calvo, M. P.; Sanz-Serna, J. M., Canonical B-series, Numer. Math., 67, 2, 161-175 (1994) · Zbl 0791.65049
[14] Cardone, A.; Jackiewicz, Z.; Sandu, A.; Zhang, H., Extrapolated IMEX Runge-Kutta methods, Math. Model. Anal., 19, 2, 18-43 (2014) · Zbl 1488.65174
[15] Cardone, A.; Jackiewicz, Z.; Sandu, A.; Zhang, H., Extrapolation-based implicit-explicit general linear methods, Numer. Algorithms, 65, 3, 377-399 (2014) · Zbl 1291.65217
[16] Cardone, A.; Jackiewicz, Z.; Sandu, A.; Zhang, H., Construction of highly-stable implicit-explicit general linear methods, (de Leon, M.; Feng, W.; Feng, Z.; Gomez, J. L.; Lu, X.; Martell, J. M.; Parcet, J.; Peralta-Salas, D.; Ruan, W., Dynamical Systems, Differential Equations, and Applications. Dynamical Systems, Differential Equations, and Applications, Madrid, Spain. Dynamical Systems, Differential Equations, and Applications. Dynamical Systems, Differential Equations, and Applications, Madrid, Spain, AIMS Proceedings, vol. 85 (2015)), 185-194 · Zbl 1335.65060
[17] Celledoni, Elena; Kometa, Bawfeh Kingsley, Semi-Lagrangian Runge-Kutta exponential integrators for convection dominated problems, J. Sci. Comput., 41, 1, 139-164 (Apr 2009) · Zbl 1203.65165
[18] Certaine, John, The solution of ordinary differential equations with large time constants, (Math. Meth. Digital Comp., vol. 1 (1960)), 128-132
[19] Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles, A substitution law for B-series vector fields (2005), INRIA, PhD thesis · Zbl 1201.65124
[20] Chou, Ching-Shan; Zhang, Yong-Tao; Zhao, Rui; Nie, Qing, Numerical methods for stiff reaction-diffusion systems, Discrete Contin. Dyn. Syst., Ser. B, 7, 3, 515-525 (Feb 2007) · Zbl 1125.65080
[21] Constantinescu, E. M.; Sandu, A., Extrapolated implicit-explicit time stepping, SIAM J. Sci. Comput., 31, 6, 4452-4477 (2010) · Zbl 1209.65069
[22] Dettmer, Wulf G.; Perić, Djordje, A new staggered scheme for fluid-structure interaction, Int. J. Numer. Methods Eng., 93, 1, 1-22 (Jul 2012) · Zbl 1352.74471
[23] Ehle, B. L.; Lawson, J. D., Generalized Runge-Kutta processes for stiff initial-value problems, IMA J. Appl. Math., 16, 1, 11-21 (1975) · Zbl 0308.65046
[24] Faragó, István; Geiser, Jürgen, Iterative operator-splitting methods for linear problems, Int. J. Comput. Sci. Eng., 3, 4, 255-263 (2007)
[25] Faragó, István; Gnandt, Boglárka; Havasi, Ágnes, Additive and iterative operator splitting methods and their numerical investigation, Comput. Math. Appl., 55, 10, 2266-2279 (2008) · Zbl 1142.65374
[26] Farhat, Charbel; Park, K. C.; Dubois-Pelerin, Yves, An unconditionally stable staggered algorithm for transient finite element analysis of coupled thermoelastic problems, Comput. Methods Appl. Mech. Eng., 85, 3, 349-365 (Feb 1991) · Zbl 0764.73081
[27] Fornberg, Bengt; Driscoll, Tobin A., A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys., 155, 2, 456-467 (1999) · Zbl 0937.65109
[28] Günther, M.; Sandu, A., Multirate generalized additive Runge-Kutta methods, Numer. Math., 133, 3, 497-524 (2016) · Zbl 1344.65064
[29] Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8 (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0789.65048
[30] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14 (1996), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0859.65067
[31] Hairer, Ernst; Wanner, Gerhard; Lubich, Christian, Order conditions, trees and B-series, (Geometric Numerical Integration (2006), Springer), 51-96 · Zbl 1094.65125
[32] Hersch, Joseph, Contribution à la méthode des équations aux différences, Z. Angew. Math. Phys., 9, 2, 129-180 (1958) · Zbl 0084.11401
[33] Hochbruck, M.; Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, 5, 1911-1925 (1997) · Zbl 0888.65032
[34] Hochbruck, M.; Lubich, C.; Selhofer, H., Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19, 5, 1552-1574 (1998) · Zbl 0912.65058
[35] Hochbruck, M.; Ostermann, A., Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43, 1069-1090 (2005) · Zbl 1093.65052
[36] Hochbruck, M.; Ostermann, A., Exponential integrators, Acta Numer., 19, 209-286 (2012) · Zbl 1242.65109
[37] Hochbruck, M.; Ostermann, A.; Schweitzer, J., Exponential Rosenbrock-type methods, SIAM J. Numer. Anal., 47, 786-803 (2009) · Zbl 1193.65119
[38] Hochbruck, Marlis; Lubich, Christian; Selhofer, Hubert, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19, 5, 1552-1574 (Sep 1998) · Zbl 0912.65058
[39] Issa, Raad I., Solution of the implicitly discretised fluid flow equations by operator-splitting, J. Comput. Phys., 62, 1, 40-65 (1986) · Zbl 0619.76024
[40] Karlsen, Kenneth Hvistendahl; Risebro, Nils Henrik, An operator splitting method for nonlinear convection-diffusion equations, Numer. Math., 77, 3, 365-382 (1997) · Zbl 0882.35074
[41] Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 1-2, 139-181 (2003) · Zbl 1013.65103
[42] Douglas Lawson, J., Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal., 4, 3, 372-380 (Sep 1967) · Zbl 0223.65030
[43] Li, Dongping; Cong, Yuhao; Xia, Kaifeng, Flexible exponential integration methods for large systems of differential equations, J. Appl. Math. Comput., 51, 1-2, 545-567 (Aug 2015) · Zbl 1341.65027
[44] Lie, K-A.; Vidar Haugse; Hvistendahl Karlsen, K., Dimensional splitting with front tracking and adaptive grid refinement, Numer. Methods Partial Differ. Equ., 14, 5, 627-648 (1998) · Zbl 0923.65061
[45] Loffeld, J.; Tokman, M., Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs, J. Comput. Appl. Math., 241, 45-67 (Mar 2013) · Zbl 1258.65067
[46] Lorenz, Edward N., Predictability – a problem partly solved, (Palmer, Tim; Hagedorn, Renate, Predictability of Weather and Climate (1996), Cambridge University Press (CUP)), 40-58
[47] Thai Luan, Vu; Ostermann, Alexander, Exponential B-series: the stiff case, SIAM J. Numer. Anal., 51, 6, 3431-3445 (2013) · Zbl 1285.65043
[48] Thai Luan, Vu; Ostermann, Alexander, Exponential Rosenbrock methods of order five - construction, analysis and numerical comparisons, J. Comput. Appl. Math., 255, 417-431 (2014) · Zbl 1291.65201
[49] Luan, Vu Thai; Tokman, Mayya; Rainwater, Greg, Preconditioned implicit-exponential integrators (IMEXP) for stiff PDEs, J. Comput. Phys., 335, 846-864 (Apr 2017) · Zbl 1378.65140
[50] MacNamara, Shev; Strang, Gilbert, Operator Splitting, 95-114 (2016), Springer International Publishing: Springer International Publishing Cham · Zbl 1372.65236
[51] Mahara, Hitoshi; Suematsu, Nobuhiko J.; Yamaguchi, Tomohiko; Ohgane, Kunishige; Nishiura, Yasumasa; Shimomura, Masatsugu, Three-variable reversible Gray-Scott model, J. Chem. Phys., 121, 18, 8968-8972 (2004)
[52] Mahara, Hitoshi; Yamaguchi, Tomohiko; Shimomura, Masatsugu, Entropy production in a two-dimensional reversible Gray-Scott system, Chaos, 15, 4, 8968 (2005) · Zbl 1144.37380
[53] McLachlan, Robert I.; Reinout, G.; Quispel, W., Splitting methods, Acta Numer., 11, 341-434 (Jan 2002) · Zbl 1105.65341
[54] Borislav V. Minchev, Will M. Wright, A review of exponential integrators for first order semi-linear problems, 2005.
[55] Nakamura, Takashi; Tanaka, Ryotaro; Yabe, Takashi; Takizawa, Kenji, Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, J. Comput. Phys., 174, 1, 171-207 (2001) · Zbl 0995.65094
[56] Narayanamurthi, M.; Romer, U.; Sandu, A., Solving parameter estimation problems with discrete adjoint exponential integrators, Optim. Methods Softw., 33, 4-6, 750-770 (2018) · Zbl 1401.49050
[57] Narayanamurthi, M.; Tranquilli, P.; Sandu, A.; Tokman, M., EPIRK-W and EPIRK-K time discretization methods, J. Sci. Comput., 78, 1, 167-201 (2019) · Zbl 1410.65255
[58] Narayanamurthi, Mahesh; Sandu, Adrian, Partitioned exponential methods for coupled multiphysics systems (Aug 2019), arXiv e-prints
[59] Nie, Qing; Zhang, Yong-Tao; Zhao, Rui, Efficient semi-implicit schemes for stiff systems, J. Comput. Phys., 214, 2, 521-537 (May 2006) · Zbl 1089.65094
[60] Niesen, Jitse; Wright, Will M., Algorithm 919: a Krylov subspace algorithm for evaluating the φ-functions appearing in exponential integrators, ACM Trans. Math. Softw., 38, 3, 22:1-22:19 (2012) · Zbl 1365.65185
[61] Pearson, John E., Complex patterns in a simple system (1993), Technical Report 5118
[62] Rainwater, G.; Tokman, Mayya, A new class of split exponential propagation iterative methods of Runge-Kutta type (sEPIRK) for semilinear systems of ODEs, J. Comput. Phys., 269, 40-60 (2014) · Zbl 1349.65227
[63] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29, 1, 209-228 (Feb 1992) · Zbl 0749.65030
[64] Sandu, A.; Borden, C. T., A framework for the numerical treatment of aerosol dynamics, Appl. Numer. Math., 45, 4, 475-497 (2003) · Zbl 1022.76040
[65] Sandu, A.; Günther, M., A generalized-structure approach to additive Runge-Kutta methods, SIAM J. Numer. Anal., 53, 1, 17-42 (2015) · Zbl 1327.65132
[66] Sidje, Roger B., Expokit: a software package for computing matrix exponentials, ACM Trans. Math. Softw., 24, 1, 130-156 (Mar 1998) · Zbl 0917.65063
[67] Smith, Barry; Bjorstad, Petter; Gropp, William, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (2004), Cambridge University Press · Zbl 0857.65126
[68] Sportisse, Bruno, An analysis of operator splitting techniques in the stiff case, J. Comput. Phys., 161, 1, 140-168 (2000) · Zbl 0953.65062
[69] Steihaug, Trond; Wolfbrandt, Arne, An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations, Math. Comput., 33, 146 (May 1979), 521-521 · Zbl 0451.65055
[70] Tang, Y.; Carmichael, G. R.; Horowitz, L. W.; Uno, I.; Woo, J. H.; Streets, D. G.; Dabdub, D.; Kurata, G.; Sandu, A.; Allan, J.; Atlas, E.; Flocke, F.; Huey, L. G.; Jakoubek, R. O.; Millet, D. B.; Parrish, D. D.; Quinn, P. K.; Roberts, J. M.; Ryerson, T. B.; Williams, E.; Nowak, J. B.; Worsnop, D.; Goldstein, A.; Donnelly, S.; Schauffler, S.; Stroud, V.; Johnson, K.; Avery, M. A.; Singh, H. B.; Apel, E. C., Multi-scale simulations of tropospheric chemistry in the Eastern Pacific and U.S. West coast during spring 2002, J. Geophys. Res., Atmos., 109, D23 (2004)
[71] Tokman, M., Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, J. Comput. Phys., 213, 2, 748-776 (2006) · Zbl 1089.65063
[72] Tokman, M., A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK), J. Comput. Phys., 230, 8762-8778 (2011) · Zbl 1370.65035
[73] Tokman, Mayya; Loffeld, John; Tranquilli, Paul, New adaptive exponential propagation iterative methods of Runge-Kutta type, SIAM J. Sci. Comput., 34, 5, A2650-A2669 (2012) · Zbl 1259.65121
[74] Toselli, Andrea; Widlund, Olof B., Domain Decomposition Methods — Algorithms and Theory (2005), Springer: Springer Berlin, Heidelberg · Zbl 1069.65138
[75] Tranquilli, P.; Sandu, A., Exponential-Krylov methods for ordinary differential equations, J. Comput. Phys., 278, 31-46 (2014) · Zbl 1349.65228
[76] Tranquilli, P.; Sandu, A., Rosenbrock-Krylov methods for large systems of differential equations, SIAM J. Sci. Comput., 36, 3, A1313-A1338 (2014) · Zbl 1320.65108
[77] Verwer, J. G.; Sommeijer, B. P., An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations, SIAM J. Sci. Comput., 25, 5, 1824-1835 (2004) · Zbl 1061.65090
[78] Verwer, Jan G., Contractivity of locally one-dimensional splitting methods, Numer. Math., 44, 2, 247-259 (1984) · Zbl 0539.65082
[79] Zhang, H.; Sandu, A., A second-order diagonally-implicit-explicit multi-stage integration method, (Proceedings of the International Conference on Computational Science. Proceedings of the International Conference on Computational Science, ICCS 2012, vol. 9 (April 2012)), 1039-1046
[80] Zhang, H.; Sandu, A.; Blaise, S., Partitioned and implicit-explicit general linear methods for ordinary differential equations, J. Sci. Comput., 61, 1, 119-144 (2014) · Zbl 1308.65122
[81] Zhang, H.; Sandu, A.; Blaise, S., High order implicit-explicit general linear methods with optimized stability regions, SIAM J. Sci. Comput., 38, 3, A1430-A1453 (2016) · Zbl 1337.65008
[82] Zhao, Su; Ovadia, Jeremy; Liu, Xinfeng; Zhang, Yong-Tao; Nie, Qing, Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems, J. Comput. Phys., 230, 15, 5996-6009 (Jul 2011) · Zbl 1220.65120
[83] Zharovsky, E.; Sandu, A.; Zhang, H., A class of IMEX two-step Runge-Kutta methods, SIAM J. Numer. Anal., 53, 1, 321-341 (2015) · Zbl 1327.65133
[84] Zienkiewicz, O. C.; Paul, D. K.; Chan, A. H.C., Unconditionally stable staggered solution procedures for soil-pore fluid interaction problems, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 25, 5, 233 (Oct 1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.