Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914148159324160 |
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| author | Lampert, Joshua Ranocha, Hendrik |
| author_facet | Lampert, Joshua Ranocha, Hendrik |
| contents | We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Svärd and Kalisch (2025) with enhanced dispersive behavior. Both models share the property of being conservative in terms of a nonlinear invariant, often interpreted as energy. This property is preserved exactly in our novel semidiscretizations. To obtain fully-discrete energy-stable schemes, we employ the relaxation method. Our novel methods generalize energy-conserving methods for the BBM-BBM system to variable bathymetries. Compared to the low-order, energy-dissipative finite volume method proposed by Svärd and Kalisch, our schemes are arbitrary high-order accurate, energy-conservative or -stable, can deal with periodic and reflecting boundary conditions, and can be any method within the framework of summation-by-parts operators including finite difference and finite element schemes. We present improved numerical properties of our methods in some test cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations Lampert, Joshua Ranocha, Hendrik Numerical Analysis 65M06, 65M20, 65M70 We use the general framework of summation-by-parts operators to construct conservative, energy-stable, and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Svärd and Kalisch (2025) with enhanced dispersive behavior. Both models share the property of being conservative in terms of a nonlinear invariant, often interpreted as energy. This property is preserved exactly in our novel semidiscretizations. To obtain fully-discrete energy-stable schemes, we employ the relaxation method. Our novel methods generalize energy-conserving methods for the BBM-BBM system to variable bathymetries. Compared to the low-order, energy-dissipative finite volume method proposed by Svärd and Kalisch, our schemes are arbitrary high-order accurate, energy-conservative or -stable, can deal with periodic and reflecting boundary conditions, and can be any method within the framework of summation-by-parts operators including finite difference and finite element schemes. We present improved numerical properties of our methods in some test cases. |
| title | Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations |
| topic | Numerical Analysis 65M06, 65M20, 65M70 |
| url | https://arxiv.org/abs/2402.16669 |