High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations

doi:10.1007/s42967-020-00110-5

Abstract

Abstract: We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great fexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-difusion equation and in the setting of strong stability preserving (SSP) methods. Our fndings are demonstrated on several examples, including nonlinear reaction-difusion and convection-difusion problems.

Key words: Semi-implicit methods, Implicit-explicit methods, Multistep methods, Strong stability preserving, High order accuracy

CLC Number:

Giacomo Albi, Lorenzo Pareschi. High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 701-718.

TrendMD

References

1. Akrivis, G.: Implicit-explicit multistep methods for nonlinear parabolic equations. Math. Comput. 82, 45–68 (2012)
2. Akrivis, G., Crouzeix, M., Makridakis, C.: Implicit-explicit multistep methods for quasilinear parabolic equations. Numer. Math. 82, 521–541 (1999)
3. Albi, G., Dimarco, G., Pareschi, L.: Implicit-explicit multistep methods for hyperbolic systems with multiscale relaxation. arXiv:1904.03865 (2019)
4. Albi, G., Herty, M., Pareschi, L.: Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems. Appl. Math. Comput. 34, 460–477 (2019)
5. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial diferential equations. Appl. Numer. Math. 25, 151–167 (1997)
6. Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial diferential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)
7. Bertozzi, A.L.: The mathematics of moving contact lines in thin liquid flms. Not. AMS 45, 689–697 (1998)
8. Boscarino, S., Bürger, R., Mulet, P., Russo, G., Villada, L.M.: On linearly implicit IMEX Runge-Kutta methods for degenerate convection-difusion problems modeling polydisperse sedimentation. Bull. Braz. Math. Soc. New Ser. 47(1), 171–185 (2016)
9. Boscarino, S., Filbet, F., Russo, G.: High order semi-implicit schemes for time dependent partial differential equations. J. Sci. Comput. 68, 975–1001 (2016)
10. Boscarino, S., Pareschi, L.: On the asymptotic properties of IMEX Runge-Kutta schemes for hyperbolic balance laws. J. Comput. Appl. Math. 316, 60–73 (2017)
11. Boscarino, S., Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the difusion limit. SIAM J. Sci. Comput. 35, A22–A51 (2011)
12. Boscarino, S., Pareschi, L., Russo, G.: A unifed IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation. SIAM J. Numer. Anal. 55(4), 2085–2109 (2017)
13. Carpenter, M.H., Kennedy, C.A.: Additive Runge-Kutta schemes for convection-difusion-reaction equations. Appl. Numer. Math. 44(1/2), 139–181 (2003)
14. Dimarco, G., Pareschi, L.: Asymptotic-preserving IMEX Runge-Kutta methods for nonlinear kinetic equations. SIAM J. Numer. Anal. 51, 1064–1087 (2013)
15. Dimarco, G., Pareschi, L.: Implicit-explicit linear multistep methods for stif kinetic equations. SIAM J. Numer. Anal. 55(2), 664–690 (2017)
16. Dimarco, G., Pareschi, L.: Numerical methods for kinetic equations. Acta Numerica 23, 369–520 (2014)
17. Filbet, F., Jin, S.: A class of asymptotic preserving schemes for kinetic equations and related problems with stif sources. J. Comput. Phys. 229, 7625–7648 (2010)
18. Frank, J., Hundsdorfer, W., Verwer, J.G.: On the stability of implicit-explicit linear multistep methods. Appl. Numer. Math. 25, 193–205 (1997)
19. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
20. Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225, 2016–2042 (2007)
21. Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)
22. Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations. Computational Mathematics. Springer, Berlin (2003)
23. Hairer, E., Wanner, G.: Solving Ordinary Diferential Equation Ⅱ: Stif and Diferential Algebraic Problems. Computational Mathematics, vol. 14. Springer, Berlin (1991). Second revised edition 1996
24. Jin, S.: Efcient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)
25. Naldi, G., Pareschi, L., Toscani, G.: Relaxation schemes for PDEs and applications to degenerate difusion problems. Surv. Math. Ind. (Eur. J. Appl. Math.) 10, 315–343 (2002)
26. Pearson, J.E.: Complex patterns in a simple system. Science 261(5118), 189–192 (1993)
27. Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxations. J. Sci. Comput. 25, 129–155 (2005)
28. Pareschi, L., Russo, G., Toscani, G.: A kinetic approximation to Hele-Shaw fow. Comptes Rendus Math. Acad. Sci. Ser. I 338(2), 178–182 (2004)
29. Pareschi, L., Zanella, M.: Structure preserving schemes for nonlinear Fokker-Planck equations and applications. J. Sci. Comput. 74(3), 1575–1600 (2018)
30. Rosales, R.R., Seibold, B., Shirokof, D., Zhou, D.: Unconditional stability for multistep IMEX schemes: theory. SIAM J. Numer. Anal. 55(5), 2336–2360 (2017)
31. Seibold, B., Shirokof, D., Zhou, D.: Unconditional stability for multistep IMEX schemes: practice. J. Comput. Phys. 376, 295–321 (2019)
32. Zhang, K., Wong, J.C.F., Zhang, R.: Second-order implicit-explicit scheme for the Gray-Scott model. J. Comput. Appl. Math. 213, 559–581 (2008)

[1]	Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian. A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 31-55.
[2]	Yubo Yang, Fanhai Zeng. Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 621-637.
[3]	Zachary Grant, Sigal Gottlieb, David C. Seal. A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 21-59.