Volume 29, Issue 4
A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

Hendrik Ranocha, Dimitrios MitsotakisDavid I. Ketcheson

Commun. Comput. Phys., 29 (2021), pp. 979-1029.

Published online: 2021-02

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  • 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.

  • Keywords

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

  • AMS Subject Headings

65M12, 65M70, 65M06, 65M60, 65M20, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-979, author = {Ranocha , Hendrik and Mitsotakis , Dimitrios and I. Ketcheson , David}, title = {A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {4}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0119}, url = {http://global-sci.org/intro/article_detail/cicp/18643.html} }
TY - JOUR T1 - A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations AU - Ranocha , Hendrik AU - Mitsotakis , Dimitrios AU - I. Ketcheson , David JO - Communications in Computational Physics VL - 4 SP - 979 EP - 1029 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0119 UR - https://global-sci.org/intro/article_detail/cicp/18643.html KW - Invariant conservation, summation by parts, finite difference methods, Galerkin methods, relaxation schemes. AB -

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.

Hendrik Ranocha, Dimitrios Mitsotakis & David I. Ketcheson. (2021). A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations. Communications in Computational Physics. 29 (4). 979-1029. doi:10.4208/cicp.OA-2020-0119
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