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arXiv · 2402.16669

Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations

Joshua Lampert·Hendrik Ranocha
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Abstract

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 Sv\"ard 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 Sv\"ard 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.

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Joshua Lampert, Hendrik Ranocha. 2024-02-26. Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations. https://doi.org/10.1007/s44207-025-00006-3

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