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A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations
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Year:    2021

Author:    Hendrik Ranocha, Dimitrios Mitsotakis, David I. Ketcheson

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 979–1029

Abstract

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0119

Communications in Computational Physics, Vol. 29 (2021), Iss. 4 : pp. 979–1029

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    51

Keywords:    Invariant conservation summation by parts finite difference methods Galerkin methods relaxation schemes.

Author Details

Hendrik Ranocha

Dimitrios Mitsotakis

David I. Ketcheson

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  17. Efficient Implementation of Modern Entropy Stable and Kinetic Energy Preserving Discontinuous Galerkin Methods for Conservation Laws

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