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A fast spectral algorithm for nonlinear wave equations with linear dispersion. (English) Zbl 0937.65109

The authors presented an easily implemented time stepping strategy for spatially spectral numerical solution to a wide range of nonlinear wave equations. The methods combine Adams-Bashforth and Adams-Moulton methods for the nonlinear and stiff linear parts, respectively, with the novel feature that different methods are used in different wavenumber ranges. The result combines high temporal accuracy with good stability properties. Numerical tests conducted on the Korteweg-de Vries and nonlinear Schrödinger equations show that the new approach is computationally more effective than other currently available methods.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)

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References:

[1] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-depdendent partial differential equations, Appl. Num. Math., 25, 151 (1997) · Zbl 0896.65061
[2] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T.R., Implicit-explicit methods for time-depdendent partial differential equations, SIAM J. Numer. Anal., 32, 797 (1995) · Zbl 0841.65081
[3] Calvo, M. P.; de Frutos, J.; Novo, J., Linearly Implicit Runge-Kutta Methods for Advection-Reaction-Diffusion Equations (1999) · Zbl 0983.65106
[4] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamics (1988) · Zbl 0658.76001
[5] Chan, T. F.; Kerkhoven, T., Fourier methods with extended stability intervals for the Korteweg-de Vries equation, SIAM J. Numer. Anal., 22, 441 (1985) · Zbl 0571.65082
[6] Fornberg, B., A Practical Guide to Pseudospectral Methods (1996) · Zbl 0844.65084
[7] Garcı́a-Archilla, B., Some practical experience with the time integration of dissipative equations, J. Comput. Phys, 122, 25 (1995) · Zbl 0854.65078
[8] M. Ghrist, High-order Finite Difference Schemes for Wave Equations, Ph.D. thesis, University of Colorado, Boulder, CO, in preparation.; M. Ghrist, High-order Finite Difference Schemes for Wave Equations, Ph.D. thesis, University of Colorado, Boulder, CO, in preparation.
[9] M. Ghrist, T. A. Driscoll, and, B. Fornberg, Staggered time stepping methods for wave equations, submitted.; M. Ghrist, T. A. Driscoll, and, B. Fornberg, Staggered time stepping methods for wave equations, submitted. · Zbl 0973.65070
[10] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time Dependent Problems and Difference Methods (1995) · Zbl 0843.65061
[11] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (1993) · Zbl 0789.65048
[12] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (1996) · Zbl 0859.65067
[13] P. A. Milewski, and, E. G. Tabak, A pseudo-spectral procedure for the solution of nonlinear wave equations with examples from free-surface flows, SIAM J. Sci. Comput, in press.; P. A. Milewski, and, E. G. Tabak, A pseudo-spectral procedure for the solution of nonlinear wave equations with examples from free-surface flows, SIAM J. Sci. Comput, in press. · Zbl 0953.65073
[14] Taflove, A., Computational Electrodynamics: The Finite-Difference Time-Domain Method (1995) · Zbl 0840.65126
[15] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical, J. Comput. Phys., 55, 192 (1984) · Zbl 0541.65081
[16] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, J. Comput. Phys., 55, 203 (1984) · Zbl 0541.65082
[17] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation, J. Comput. Phys., 55, 231 (1984) · Zbl 0541.65083
[18] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 262 (1990)
[19] Zabusky, N. J.; Kruskal, M. D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240 (1965) · Zbl 1201.35174
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