Giuseppe Orlando

Le 22 janv. 2026

Titre : An adaptive high-order DG solver for the shallow water equations with irregular bathymetry

Résumé :

We present the first step in the development of an adaptive Discontinuous Galerkin (DG) solver for the shallow water equations for coastal modelling [1]. In this work, we restrict our attention to a second order IMEX time discretization that treats the pressure gradient term explicitly, while applying an implicit method to the friction term. The IMEX scheme is based on the combination of the L-stable TR-BDF2 for the implicit part and an explicit three stages second order Runge-Kutta scheme designed to match the coupling conditions and to improve the absolute monotonicity of the scheme. This choice leads to linear systems involving only block-diagonal operators, which can be easily inverted.

We focus in particular on a robust and accurate treatment of the bathymetry, nowadays available with higher resolution than the mesh in coastal areas. This poses a series of challenges for high-order methods that work on quite coarse meshes. In particular, the mesh may not be aligned to large bathymetric gradients or jumps and the DG method should be therefore able to handle large gradients or even discontinuities within an element or along an edge. For this reason, we choose as prognostic variable the free-surface elevation, which is smooth for subcritical regimes, and for which we employ a finite element representation. The bathymetry at the quadrature node is directly evaluated from the reference data without any local modification, smoothing, interpolation or hydrostatic reconstruction [1]. Moreover, this choice guarantees automatic verification of the well-balancing property with respect to the lake-at-rest. We show the robustness of this approach in presence of a realistic bathymetry while, for the opposite choice of the water depth as prognostic variable, a smoothing or a TVD limiting of the bathymetry is necessary to avoid the Gibbs phenomenon [1].

The spatial discretization is based on a high-order DG method as implemented in the deal.II library. In this framework, we test non-conforming meshes with static and dynamic Adaptive Mesh Refinement (AMR) so as to simulate the tidal circulations in the complex coastal environment of the Venice Lagoon. We show that dynamic AMR allows to resolve small-scale structures that are absent in the static runs. Finally, preliminary results for the correct treatment of wetting-drying will be also addressed.

This work is based on a collaboration with Luca Arpaia, Christian Ferrarin (Istituto di Scienze Marine, CNR, Italy), and Luca Bonaventura (Politecnico di Milano).

[1] L. Arpaia, G. Orlando, C. Ferrarin, and L. Bonaventura. A high-order matrix-free adaptive solver for the shallow water equations with irregular bathymetry, 2025. https://arxiv.org/abs/2505.18743.