Exponential time differencing for stiff systems. (English) Zbl 1005.65069

Summary: We develop a class of numerical methods for stiff systems, based on the method of exponential time differencing. We describe schemes with second- and higher-order accuracy, introduce new Runge-Kutta versions of these schemes, and extend the method to show how it may be applied to systems whose linear part is nondiagonal. We test the method against other common schemes, including integrating factor and linearly implicit methods, and show how it is more accurate in a number of applications. We apply the method to both dissipative and dispersive partial differential equations, after illustrating its behavior using forced ordinary differential equations with stiff linear parts.


65L05 Numerical methods for initial value problems involving ordinary differential equations
35K30 Initial value problems for higher-order parabolic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L20 Stability and convergence of numerical methods for ordinary differential equations


Full Text: DOI Link


[1] Beylkin, G.; Keiser, J. M.; Vozovoi, L., A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys., 147, 362 (1998) · Zbl 0924.65089
[2] Boyd, J. P., Eight definitions of the slow manifold: Seiches, pseudoseiches and exponential smallness, Dyn. Atmos. Oceans, 22, 49 (1995)
[3] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover New York · Zbl 0987.65122
[4] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0658.76001
[5] Fornberg, B., A Practical Guide to Pseudospectral Methods (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[6] Fornberg, B.; Driscoll, T. A., A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys., 155, 456 (1999) · Zbl 0937.65109
[7] Garcı́a-Archilla, B., Some practical experience with the time integration of dissipative equations, J. Comput. Phys., 122, 25 (1995) · Zbl 0854.65078
[8] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations (1962), Wiley: Wiley New York · Zbl 0112.34901
[9] Holland, R., Finite-difference time-domain (FDTD) analysis of magnetic diffusion, IEEE Trans. Electromagn. Compat., 36, 32 (1994)
[10] Iserles, A., A First Course in the Numerical Analysis of Differential Equations (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[11] King, J. R.; Cox, S. M., Asymptotic analysis of the steady-state and time-dependent Berman problem, J. Eng. Math., 39, 87 (2001) · Zbl 1009.76023
[12] Kuramoto, Y.; Tsuzuki, T., Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55, 356 (1976)
[13] Marion, M.; Temam, R., Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26, 1139 (1989) · Zbl 0683.65083
[14] Matthews, P. C.; Cox, S. M., Pattern formation with a conservation law, Nonlinearity, 13, 1293 (2000) · Zbl 0960.35007
[15] Matthews, P. C.; Cox, S. M., One-dimensional pattern formation with Galilean invariance near a stationary bifurcation, Phys. Rev. E, 62, R1473 (2000)
[16] Milewski, P. A.; Tabak, E., A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows, SIAM J. Sci. Comp., 21, 1102 (1999) · Zbl 0953.65073
[17] Petropoulos, P. G., Analysis of exponential time-differencing for FDTD in lossy dielectrics, IEEE Trans. Antennas Propagation, 45, 1054 (1997)
[18] Schuster, C.; Christ, A.; Fichtner, W., Review of FDTD time-stepping for efficient simulation of electric conductive media, Microwave Optical Technol. Lett., 25, 16 (2000)
[19] Taflove, A., Computational Electrodynamics: The Finite-Difference Time-Domain Method (1995), Artech House Antenna Library Artech House: Artech House Antenna Library Artech House London · Zbl 0840.65126
[22] Zhu, J. Z.; Chen, L.-Q.; Shen, J.; Tikare, V., Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method, Phys. Rev. E, 60, 3564 (1999)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.