A fully well-balanced hydrodynamic reconstruction
Christophe Berthon, Victor Michel-Dansac
TL;DR
This work advances shallow water simulations over uneven topography by extending the hydrostatic reconstruction to a hydrodynamic reconstruction that exactly preserves moving steady states and the lake-at-rest solution, without solving nonlinear Bernoulli equations. The key ideas include a linearized interface reconstruction governed by a perturbation function $\mathcal{H}$ that satisfies specific consistency, well-balanced, and dry/wet-dynamics properties, plus a high-order extension that uses a steady-state detector to maintain well-balancedness without nonlinear solves. Comprehensive numerical experiments demonstrate consistency, accuracy, and robust handling of wet/dry transitions, dam-breaks, and stationary contacts, while highlighting the method’s limitations near critical points and its dependence on the choice of $\mathcal{H}$. The proposed framework delivers a versatile, computationally efficient approach for reliable shallow water simulations on complex topographies, with potential for further refinement of $\mathcal{H}$ and enhanced performance at Fr$^2\approx1$.
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.