Exponential Time Differencing for Stiff Systems

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.

[1]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

[2]  Y. Kuramoto,et al.  Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium , 1976 .

[3]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[4]  J. Boyd.  Chebyshev and Fourier Spectral Methods , 1989 .

[5]  R. Temam,et al.  Nonlinear Galerkin methods , 1989 .

[6]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[7]  R. Holland,et al.  Finite-difference time-domain (FDTD) analysis of magnetic diffusion , 1994 .

[8]  Bosco García-Archilla.  Some Practical Experience with the Time Integration of Dissipative Equations , 1995 .

[9]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[10]  Eight definitions of the slow manifold: seiches, pseudoseiches and exponential smallness , 1995 .

[11]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[12]  Analysis of exponential time-differencing for FDTD in lossy dielectrics , 1997 .

[13]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[14]  G. Akrivis.  A First Course In The Numerical Analysis Of Differential Equations [Book News & Reviews] , 1998, IEEE Computational Science and Engineering.

[15]  Esteban G. Tabak,et al.  A PseudoSpectral Procedure for the Solution of Nonlinear Wave Equations with Examples from Free-Surface Flows , 1999, SIAM J. Sci. Comput..

[16]  T. Driscoll,et al.  Regular Article: A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion , 1999 .

[17]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  Matthews,et al.  One-dimensional pattern formation with galilean invariance near a stationary bifurcation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Wolfgang Fichtner,et al.  Review of FDTD time-stepping schemes for efficient simulation of electric conductive media , 2000 .

[20]  Stephen M. Cox,et al.  Pattern formation with a conservation law , 2000, nlin/0006002.

[21]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[22]  Asymptotic analysis of the steady-state and time-dependent Berman problem , 2001 .