GPU-Accelerated Energy-Conserving Methods for the Two-Dimensional Hyperbolized Serre-Green-Naghdi Equations
This work provides efficient, energy-conserving simulations for coastal and ocean engineering applications, though it is an incremental improvement over existing numerical methods for hyperbolic formulations.
The authors developed energy-conserving numerical methods for the two-dimensional hyperbolized Serre-Green-Naghdi equations, achieving substantial speedups on GPUs while avoiding costly elliptic operator inversions. The methods conserve total water mass and energy, validated through convergence studies and comparisons to analytical and experimental results.
We develop energy-conserving numerical methods for a two-dimensional hyperbolic approximation of the Serre-Green-Naghdi equations with variable bathymetry and either periodic or 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 semi-discretizations 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.