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.

Keywords

Bibliographic record

BibTeX Citation

@article{Chehab2019timerelaxed,
  author = {Chehab, J.-P. and Dutykh, D.},
  title = {On time relaxed schemes and formulations for dispersive wave equations},
  journal = {AIMS Mathematics},
  year = {2019},
  volume = {4},
  number = {2},
  pages = {254--278},
  doi = {10.3934/math.2019.2.254},
  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.},
  keywords = {relaxation, water waves, high order derivatives, numerical simulation, nonlinear waves, Korteweg-de Vries equation, evolution problem, dispersive wave equations, first order derivatives, shallow water flows, stiff dispersive PDEs, relaxed system formulations, quasi-compressibility, stability restrictions}
}