High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products

TL;DR

This paper introduces high-order well-balanced WENO finite volume schemes for shallow water moment equations with non-conservative products, effectively preserving steady states without requiring their analytical forms.

Contribution

It develops a flux globalization approach-based method that handles non-conservative products and preserves steady states in complex hyperbolic systems without explicit steady state knowledge.

Findings

Optimal convergence demonstrated in numerical tests

Significant error reduction in steady state solutions

Flexible method applicable to general equations without prior steady state info

Abstract

Shallow water moment equations are reduced-order models for free-surface flows that allow to represent vertical variations of the velocity profile at the expense of additional evolution equations for a number of additional variables, so called moments. This introduces non-linear non-conservative products in the system, which make the analytical characterization of steady states much harder if not impossible. The lack of analytical steady states poses a challenge for the design of well-balanced schemes, which aim at preserving such steady states as crucial in many applications. In this work, we present a family of fully well-balanced, high-order WENO finite volume methods for general hyperbolic balance laws with non-conservative products like the shallow water moment equations, for which no analytical steady states are available. The schemes are based on the flux globalization approach, in which both source terms and non-conservative products are integrated with a tailored high order quadrature in the divergence term. The resulting global flux is then reconstructed instead of the conservative variables to preserve all steady states. Numerical tests show the optimal convergence of the method and a significant error reduction for steady state solutions. Furthermore, we provide a numerical comparison of perturbed steady states for different families of shallow water moment equations, which illustrates the flexibility of our method that is valid for general equations without prior knowledge of steady states.