New Numerical Schemes Keep Dispersive Wave Simulations Energy-Conserving

Numerical simulations of ocean waves often leak energy: as a computer grinds through the equations that describe moving water, small errors accumulate and the simulated sea slowly gains or loses energy that the real physics would conserve. A new study addresses this problem directly for two modern dispersive shallow-water models, presenting numerical methods that provably preserve the energy-like quantity at the heart of these equations—exactly, not approximately. The work, published as an open-access research article in the Springer series Computational Science and Engineering on 10 November 2025, was carried out by Joshua Lampert and Hendrik Ranocha.

The two models in question build on the classical shallow water equations, a widely used description of water waves whose wavelength is large compared with the water depth. These equations describe the water height and velocity in one spatial dimension and are the workhorse for phenomena such as tsunami propagation or coastal inundation. But they have a fundamental limitation: they contain no dispersion, the physical effect that makes waves of different wavelengths travel at different speeds. For shorter wavelengths in deeper water—for example, waves generated by submarine landslides—dispersion becomes essential, and the shallow water equations simply cannot capture it.

Dispersion can be added through higher-order derivative terms, and many established models do exactly that: the formulations of Peregrine, of Madsen and Sørensen, and the related Nwogu system are prominent examples. According to the authors, however, none of these classical dispersive models conserves the mathematical entropy function—the nonlinear invariant that, for the shallow water equations, coincides with the physical total energy of potential plus kinetic energy. The two models chosen for the new study do. The first is a variant of the coupled Benjamin-Bona-Mahony, or BBM-BBM, equations extended to variable bottom topography. The second is a dispersive system proposed recently by Svärd and Kalisch, in which tunable coefficients weight third-order derivative terms so that the model’s dispersion relation can be optimized to match that of the full Euler equations of water waves.

Both models share a special structural property: they conserve a nonlinear invariant, often interpreted as energy, in exact analogy to the entropy structure of the classical shallow water equations. For the BBM-BBM system, the conserved energy takes the same form as the shallow-water energy—a quadratic functional combining the squared total water height and the kinetic contribution of the velocity. For the Svärd-Kalisch model, the conserved quantity is a modified entropy that augments the usual energy with a term involving the spatial gradient of the velocity. Preserving these invariants numerically is the central goal of the new work, because structure-preserving discretizations typically yield physically more accurate solutions and improved stability.

The mathematical machinery behind the new methods is the framework of summation-by-parts operators, usually abbreviated SBP. These discrete derivative operators are constructed so that they mimic integration by parts exactly at the discrete level, mirroring the identity that underlies the continuous energy-conservation proofs. Combined with so-called split forms—which discretely imitate the product and chain rule—SBP operators allow a systematic route to entropy-conservative and entropy-dissipative semidiscretizations, meaning schemes discrete in space but continuous in time. The framework is broad: finite difference, finite element, discontinuous Galerkin spectral element, and flux reconstruction methods can all be formulated within it, so the new schemes are not tied to one spatial discretization technology.

The theoretical core of the paper consists of new semidiscretizations for both models together with conservation theorems. For the BBM-BBM equations with variable bathymetry, the authors generalize previously known energy-conserving schemes, which had been restricted to constant bottom topography, to the variable case. The theorems show that the schemes conserve the discrete total mass and total velocity, conserve the discrete energy when the second-derivative operator is chosen appropriately, and are well-balanced—meaning they exactly preserve the important lake-at-rest steady state in which the water surface is flat and the velocity vanishes, even over a sloping or discontinuous bottom. Upwind variants of the schemes are also constructed, replacing wide-stencil central operators with pairs of one-sided operators, and analogous results are proven for reflecting, solid-wall boundary conditions.

For the Svärd-Kalisch model, the new semidiscretizations preserve the discrete total modified entropy exactly and are likewise well-balanced; an upwind variant is entropy-stable, meaning the modified entropy can only decrease. The authors contrast these schemes with the low-order, entropy-dissipative finite volume method originally proposed by Svärd and Kalisch: the new schemes are arbitrarily high-order accurate, can handle both periodic and reflecting boundary conditions, and achieve entropy stability without introducing artificial diffusion. To carry the conservation properties from the semidiscrete to the fully discrete level—discrete in both space and time—the authors employ relaxation Runge-Kutta methods, which slightly adjust the time step so that the conserved or dissipated functional is preserved exactly at every step.

The numerical experiments support the theory. In convergence tests against an analytical solitary-wave solution of the BBM-BBM equations, the observed order of convergence matched the order of accuracy of the chosen SBP operators. In a long-time run over fifty wave periods, the total energy grew linearly without relaxation but remained constant to machine precision when the relaxation Runge-Kutta method was used, and the relaxation solution tracked the analytical solitary wave accurately for a much longer time. Lake-at-rest tests with a discontinuous bathymetry returned errors at the level of machine precision, verifying the well-balanced property, and manufactured-solution tests with variable bottom topography confirmed convergence for both model systems.

The study also probes how well the two models themselves reproduce dispersive wave propagation. Simulating traveling-wave solutions of the linearized Euler equations at several wavenumbers showed that for long waves both models perform well, while for shorter waves the BBM-BBM equations accumulate significant phase errors quickly, whereas optimized parameter sets of the Svärd-Kalisch equations continue to capture both wave speed and amplitude accurately. A particularly striking validation comes from a comparison with laboratory data: the classic experiment of Dingemans, in which waves of amplitude 0.02 pass over a trapezoidal bathymetry. Using periodic central finite difference SBP operators of order four with only 512 grid points, the new Svärd-Kalisch discretization produced results comparable to those obtained by the original authors with 36,800 points in their finest mesh, and increasing the order to six with 1024 points brought the numerical solution remarkably close to the measured data. The entropy-dissipative upwind scheme captured finer details of the wave field than the conservative central variant.

The authors conclude that for the small-amplitude waves considered, the generalized Svärd-Kalisch equations yield more accurate results than the BBM-BBM equations, consistent with their linear dispersion relations, and that upwind operators improve approximations of higher-order derivatives. They also note limitations: the analysis covers one-dimensional models without wave breaking, and no general well-posedness result exists yet for the variable-coefficient Svärd-Kalisch equations. All methods have been implemented in the open-source Julia package DispersiveShallowWater.jl, and a reproducibility repository containing the complete source code for every numerical experiment accompanies the paper, allowing other researchers to verify and extend the energy-conserving machinery for their own wave simulations.

Subject of Research: Structure-preserving numerical methods for dispersive shallow water wave equations

Article Title: Structure-preserving numerical methods for two nonlinear systems of dispersive wave equations

Article References: Lampert, J., & Ranocha, H. (2025). Structure-preserving numerical methods for two nonlinear systems of dispersive wave equations. Computational Science and Engineering, 2(1), Article 2. https://doi.org/10.1007/s44207-025-00006-3

Image Credits: AI Generated

DOI: 10.1007/s44207-025-00006-3

Keywords: dispersive wave equations, shallow water equations, summation-by-parts operators, energy conservation, BBM-BBM equations, Svärd-Kalisch model, relaxation Runge-Kutta methods, well-balanced schemes, numerical analysis, variable bathymetry, entropy stability, Julia