GPU-Accelerated Energy-Conserving Methods for the Hyperbolized Serre-Green-Naghdi Equations in 2D

TL;DR

This paper introduces energy-conserving, GPU-accelerated numerical methods for a 2D hyperbolic approximation of the Serre-Green-Naghdi equations, improving computational efficiency while maintaining physical conservation laws.

Contribution

The paper develops and implements energy-conserving hyperbolic schemes using SBP operators for the SGN equations, with GPU acceleration and comprehensive validation.

Findings

Methods conserve total water mass and energy.

Achieve significant speedups on GPU architectures.

Validated through convergence and comparison with analytical and experimental data.

Abstract

We develop energy-conserving numerical methods for a two-dimensional hyperbolic approximation of the Serre-Green-Naghdi equations with variable bathymetry for both periodic and reflecting boundary conditions. The hyperbolic formulation avoids the costly inversion of an elliptic operator present in the classical model. Our schemes combine split forms with summation-by-parts (SBP) operators to construct semidiscretizations that conserve the total water mass and the total energy. We provide analytical proofs of these conservation properties and also verify them numerically. While the framework is general, our implementation focuses on second-order finite-difference SBP operators. The methods are implemented in Julia for CPU and GPU architectures (AMD and NVIDIA) and achieve substantial speedups on modern accelerators. We validate the approach through convergence studies based on solitary-wave and manufactured-solution tests, and by comparisons to analytical, experimental, and existing numerical results. All source code to reproduce our results is available online.