On time relaxed schemes and formulations for dispersive wave equations

Jean-Paul Chehab; Denys Dutykh; Jean-Paul Chehab; Denys Dutykh

doi:10.3934/math.2019.2.254

AIMS Mathematics

2019, Volume 4, Issue 2: 254-278. doi: 10.3934/math.2019.2.254

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Research article

On time relaxed schemes and formulations for dispersive wave equations

Jean-Paul Chehab ¹,
Denys Dutykh ^{2
,
,}

1.
Université de Picardie Jules Verne, LAMFA CNRS UMR 7352, 33, rue Saint-Leu, 80039 Amiens, France
2.
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

Received: 03 December 2018 Accepted: 05 March 2019 Published: 18 March 2019
MSC : Primary: 74J15; Secondary: 74S10, 74J30

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.
- dispersive wave equations,
- shallow water flows,
- relaxation,
- quasi-compressibility
Citation: Jean-Paul Chehab, Denys Dutykh. On time relaxed schemes and formulations for dispersive wave equations[J]. AIMS Mathematics, 2019, 4(2): 254-278. doi: 10.3934/math.2019.2.254

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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.

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