A composite Runge-Kutta method for the spectral solution of semilinear PDEs

Authors:
Tobin A. Driscoll
Affiliations:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware
Venue:
Journal of Computational Physics
Year:
2002

Citing 10
Cited 5

Analysis of some Krylov subspace approximations to the matrix exponential operator

SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems

Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Implicit-explicit methods for time-dependent partial differential equations

SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations

Applied Numerical Mathematics - Special issue on time integration
Symplectic Methods Based on Decompositions

SIAM Journal on Numerical Analysis
Exponential Integrators for Large Systems of Differential Equations

SIAM Journal on Scientific Computing
A new class of time discretization schemes for the solution of nonlinear PDEs

Journal of Computational Physics
A fast spectral algorithm for nonlinear wave equations with linear dispersion

Journal of Computational Physics
A PseudoSpectral Procedure for the Solution of Nonlinear Wave Equations with Examples from Free-Surface Flows

SIAM Journal on Scientific Computing
Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations

Applied Numerical Mathematics

Additive Runge-Kutta schemes for convection-diffusion-reaction equations

Applied Numerical Mathematics
An additive semi-implicit Runge--Kutta family of schemes for nonstiff systems

Applied Numerical Mathematics
Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes

Journal of Computational Physics
Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation

Journal of Computational Physics
Fourth Order Time-Stepping for Kadomtsev-Petviashvili and Davey-Stewartson Equations

SIAM Journal on Scientific Computing

Quantified Score

Hi-index	31.46

Visualization

Abstract

A new composite Runge-Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg-de Vries, nonlinear Schrödinger, Kadomtsev-Petviashvili (KP), Kuramoto-Sivashinsky (KS), Cahn-Hilliard, and others having high-order derivatives in the linear term. The method uses Fourier collocation and the classical fourth-order RK method, except for the stiff linear modes, which are treated with a linearly implicit RK method. The composite RK method is simple to implement, indifferent to the distinction between dispersive and dissipative problems, and as efficient on test problems for KS and KP as any other generally applicable method.