On time relaxed schemes and formulations for dispersive wave equations

Abstract

The numerical simulation of nonlinear dispersive waves is a central research topic of many investigations in the nonlinear wave community. Simple and robust solvers are needed for numerical studies of water waves as well. The main difficulties arise in the numerical approximation of high order derivatives and in severe stability restrictions on the time step, when explicit schemes are used. In this study we propose new relaxed system formulations which approximate the initial dispersive wave equation. However, the resulting relaxed system involves first order derivatives only and it is written in the form of an evolution problem. Thus, many standard methods can be applied to solve the relaxed problem numerically. In this article we illustrate the application of the new relaxed scheme on the classical Korteweg-de Vries equation as a prototype of stiff dispersive PDEs.

arXiv comment27 pages, 10 figures, 4 algorithms, 1 table, 36 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/

References in corpus (8)

• A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model
• Finite volume schemes for dispersive wave propagation and runup
• Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations
• On the Galerkin / finite-element method for the Serre equations
• Finite volume methods for unidirectional dispersive wave models
• Multigrid iterative algorithm using pseudo-compressibility for three-dimensional mantle convection with strongly variable viscosity
• On the Galilean invariance of some dispersive wave equations
• On the nonlinear dynamics of the traveling-wave solutions of the Serre system