A fully well-balanced hydrodynamic reconstruction

TL;DR

This paper introduces a novel, well-balanced hydrodynamic reconstruction scheme for shallow water models that preserves steady solutions, including moving states, without solving nonlinear equations, enhancing accuracy and efficiency.

Contribution

It extends hydrostatic reconstruction to exactly preserve both stationary and moving steady states without nonlinear solves, applicable to high-order schemes.

Findings

The scheme preserves steady states exactly.

It does not require solving nonlinear equations.

Numerical experiments confirm effectiveness.

Abstract

The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.