Well-Balanced Subcell Limiting for Discontinuous Galerkin Discretizations of the Shallow-Water Equations

TL;DR

This paper introduces a novel flux-differencing formulation for high-order DG methods with subcell FV limiters that preserves steady states exactly in shallow water equations with variable bottom topography.

Contribution

It proposes a new staggered flux formulation that maintains the well-balanced property at the node level during limiting, improving robustness and accuracy.

Findings

Method preserves steady states exactly during limiting.

Numerical experiments show improved stability and accuracy.

Applicable to broader nonlinear balance law systems.

Abstract

High-order discontinuous Galerkin (DG) methods equipped with subcell finite-volume (FV) limiters provide an efficient framework for the simulation of nonlinear hyperbolic balance laws featuring shocks and complex flow structures. However, for systems with non-conservative terms, the design of hybrid DG/FV schemes that simultaneously guarantee high-order accuracy, robustness, and well-balancedness remains challenging. In particular, for the shallow water equations with variable bottom topography, standard flux-differencing formulations combined with node-wise subcell limiting generally destroy the well-balanced property, even if both the underlying DG and FV methods are individually well-balanced. In this work, we propose a novel flux-differencing formulation for non-conservative systems that enables node-wise subcell limiting while preserving steady states exactly. The key idea is to construct staggered DG fluxes whose non-conservative contributions are in local-times-jump form and vanish individually at equilibrium. To achieve this, we introduce a reformulation of the shallow water equations in which the source term is proportional to the gradient of the total water height. This reformulation allows the design of staggered fluxes that preserve equilibrium locally at the node level, thereby enabling arbitrary nodal blending with low-order FV fluxes. The resulting DG/FV method is high-order accurate, robust, and exactly well-balanced under node-wise limiting. Numerical experiments, including two-dimensional dam-break configurations with wet/dry fronts and complex obstacle interactions, demonstrate the improved stability and accuracy of the proposed approach. Although this work focuses on the shallow water equations, the well-balanced hybrid DG/FV methods developed here are applicable to a broader class of nonlinear systems of balance laws.