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

TL;DR

The paper addresses accurate long-time simulation of nonlinear dispersive shallow water waves with variable bathymetry by constructing SBP-based structure-preserving discretizations for two models: BBM-BBM and Svärd–Kalisch. It develops energy- and entropy-preserving semidiscretizations and uses relaxation Runge-Kutta time stepping to achieve fully discrete energy stability, while maintaining well-balancedness for lake-at-rest states. The Svärd–Kalisch model is shown to offer superior dispersive fidelity in many regimes, and the framework supports high-order accuracy and flexible boundary conditions, including reflecting boundaries. Across numerical experiments, the methods demonstrate exact or near-exact preservation of invariants, high-order convergence, and good agreement with experimental data, highlighting their potential for reliable long-time simulations of dispersive shallow water phenomena.

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 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.

Figures (18)

• Figure 1: Water height $h$, reference total water height $\eta_0$, and bathymetry $b$. The still water depth is $D = \eta_0 - b$ and the total water height is $\eta = h + b$
• Figure 2: Dispersion relations of different models for $k\in(0, 15\pi)$
• Figure 3: Experimental order of convergence for the solitary wave solution of the BBM-BBM equation, left: baseline without relaxation, right: with relaxation
• Figure 4: Conservation of linear and nonlinear invariants for the solitary wave solution of the BBM-BBM equations, left: baseline without relaxation, right: with relaxation
• Figure 5: Evolution of $L^2$- and $L^\infty$-errors for baseline and relaxation method applied to the solitary wave solution of the BBM-BBM equations

Theorems & Definitions (22)

• theorem 2.1: Svärd and Kalisch (2023)
• proof
• definition 3.1
• definition 3.2
• example 3.3
• definition 3.4
• definition 3.5
• remark 3.6
• lemma 3.7
• proof