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

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.

Figures (16)

• Figure 1: Sketch of the variables: total water height $h(t,x,y) + b(x,y)$, water height above the bathymetry $h(t,x,y)$, and bathymetry $b(x,y)$.
• Figure 2: Convergence study for a one-dimensional solitary wave propagating in the $x$-direction and repeated in $y$. The computational domain in x is $[-30, 30]$ with $\lambda = 30\,000$. Second-order convergence is achieved for both water height $h$ and velocity $u$.
• Figure 3: Convergence study using the method of manufactured solutions on the domain $[-1, 1] \times [-1, 1]$ with $\lambda = 500$ at final time $t=1$. Left: periodic boundary conditions. Right: reflecting boundary conditions. Second-order convergence is achieved for water height $h$, velocities $u$ and $v$, and auxiliary variables $\eta$ and $w$.
• Figure 4: Numerical verification of energy conservation in the spatial semidiscretization using the wave over Gaussian test case from Section \ref{['sec:gaussian_obstacle']}. The quantity $\langle \partial_{\mathbf{q}} E, \partial_t \mathbf{q} \rangle_M$ is computed numerically. The energy time derivative fluctuates at machine precision throughout the simulation, confirming exact energy conservation of the spatial semidiscretization.
• Figure 5: Initial setup for the Dingemans experiment showing the trapezoidal bathymetry $b(x)$, initial water surface elevation $\eta(x,0)$, and locations of the six wave gauges (vertical dashed lines). The domain extends from $x = -138$ to $x = 46$ with periodic boundary conditions.

Theorems & Definitions (10)

• definition 2.1
• definition 2.2
• definition 2.3: Multidimensional SBP operator
• remark 3.1
• theorem 4.1
• proof
• remark 4.2
• theorem 4.3
• proof
• remark 4.4: Concrete SBP operators and SATs