Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for shallow water linearized moment equations

TL;DR

This paper develops high-order, well-balanced discontinuous Galerkin methods for the shallow water linearized moment equations, accurately preserving equilibrium states and handling complex velocity profiles with vertical variations.

Contribution

It introduces a novel path-conservative DG scheme that preserves both still and moving water equilibria for SWLME, addressing non-conservative terms and complex steady states.

Findings

Exact equilibrium preservation demonstrated in numerical tests

High-order accuracy maintained near complex topographies

Effective handling of vertical velocity variations

Abstract

This paper presents high-order, well-balanced, path-conservative discontinuous Galerkin (DG) methods for the shallow water linearized moment equations (SWLME), designed to preserve both still and moving water equilibrium states. Unlike the multi-layer shallow water equations, which model vertical velocity variations using multiple distinct layers, the SWLME employs a polynomial expansion of velocity profiles with up to $N$ moments. This approach enables a more detailed representation of vertical momentum transfer and complex velocity profiles while retaining hyperbolicity. However, the presence of non-conservative terms and complex steady-state structures introduces significant numerical challenges. Addressing these challenges, we develop path-conservative DG schemes grounded in the Dal Maso-LeFloch-Murat (DLM) theory for non-conservative products. Our method balances flux gradients, non-conservative terms, and source terms through equilibrium-preserving spaces. For the still water equilibrium, we reformulate the equations into a quasilinear form that eliminates source terms, inherently preserving steady states. For the moving water equilibrium, we extend the DG method by transforming conservative variables into equilibrium variables and employing linear segment paths. Theoretical analysis and numerical experiments demonstrate that the proposed methods achieve exact equilibrium preservation while maintaining high-order accuracy, even in scenarios with vertical velocity variations and complex topographies.