A high-order matrix-free adaptive solver for the shallow water equations with irregular bathymetry

TL;DR

The paper advances coastal flow simulation by introducing a high-order DG solver in deal.II with static and dynamic AMR on non-conforming meshes, using the prognostic free-surface $\zeta$ and a non-conservative hydrostatic pressure gradient to achieve automatic C-property well-balancing on irregular bathymetry. Time stepping is IMEX-RK with explicit pressure terms and implicit friction, while spatial discretization employs nodal DG with bathymetry evaluated at quadrature points to avoid smoothing. Validation spans idealized benchmarks (travelling vortex, lake-at-rest, channel flow with friction) and a Venice Lagoon case with realistic bathymetry, demonstrating accuracy, efficiency, and strong parallel performance, and highlighting the benefits of AMR for resolving sub-grid features. Future work targets mass-conserving wetting/drying, full IMEX-RK schemes with implicit pressure gradients, turbulence modeling, and comprehensive fully realistic validations.

Abstract

We present the first step in the development of an Adaptive Mesh Refinement (AMR) solver for coastal engineering applications, based on a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement both static and dynamic AMR approaches. The proposed method is automatically well-balanced, allows the use of realistic bathymetry data without any regularity assumption, and includes a consistent conservative discretization for transported chemical species. Numerical experiments on idealized benchmarks validate the proposed approach, while results obtained on realistic bathymetries and complex domains show its potential for accurate and efficient adaptive simulations of coastal flows.

Figures (19)

• Figure 1: Travelling vortex, contour plots of the free-surface elevation (left) and adapted mesh (right) at final time $T = [parse-numbers=false]{1/6}{\second}$ with $n_{max} = 5$.
• Figure 2: Travelling vortex, AMR performance metrics for different $r-$order accurate RK-DG schemes and time discretization methods. Top: absolute error on the free-surface as a function of the mesh size (red lines trace the theoretical convergence rate). Middle: absolute error as a function of the number of degrees of freedom. Bottom: wall time as a function of the error.
• Figure 3: Small perturbation of a lake-at-rest. Distorted mesh in correspondence of the bathymetry jump (red line).
• Figure 4: Small perturbation of a lake at rest, contour plots of the free-surface elevation at the final time $T = 0.48s$. Top left: RK32 and $r = 1$. Top right: RK32 and $r = 2$. Bottom left: RK32 and $r = 3$. Bottom right: reference.
• Figure 5: Small perturbation of a lake at rest, 1D cut of the approximated bathymetry along the x-axis with the distorted mesh. Red line: interpolated bathymetry with $r = 2$. Blue line: projected bathymetry with $r=2$: Black line: reference bathymetry. Points: bathymetry at the quadrature points.