A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations

TL;DR

The Saint-Venant (shallow-water) system, governing hydrostatic free-surface flows, requires accurate, physically consistent numerics that preserve nonnegative water depth and key equilibria. The authors develop a high-order finite-volume framework that combines a well-balanced spatial discretization with an unconditionally positivity-preserving time integrator based on a production-destruction reformulation and a modified Patankar deferred-correction approach. A convex WB blending between a high-order scheme and a WB discretization enables exact preservation of both static and moving equilibria, including wet/dry transitions. Numerical tests across unsteady vortex, lake-at-rest, moving equilibria, and flooding scenarios (e.g., wave over island and tsunami over obstacles) demonstrate high-order accuracy, robust positivity, and practical efficacy for flood-risk and tsunami simulations.

Abstract

In this work, we present a high-order finite volume framework for the numerical simulation of shallow water flows. The method is designed to accurately capture complex dynamics inherent in shallow water systems, particularly suited for applications such as tsunami simulations. The arbitrarily high-order framework ensures precise representation of flow behaviors, crucial for simulating phenomena characterized by rapid changes and fine-scale features. Thanks to an {\it ad-hoc} reformulation in terms of production-destruction terms, the time integration ensures positivity preservation without any time-step restrictions, a vital attribute for physical consistency, especially in scenarios where negative water depth reconstructions could lead to unrealistic results. In order to introduce the preservation of general steady equilibria dictated by the underlying balance law, the high-order reconstruction and numerical flux are blended in a convex fashion with a well-balanced approximation, which is able to provide exact preservation of both static and moving equilibria. Through numerical experiments, we demonstrate the effectiveness and robustness of the proposed approach in capturing the intricate dynamics of shallow water flows, while preserving key physical properties essential for flood simulations.

Figures (12)

• Figure 1: Sketch of the PDS structure for the control volume $\Omega_{i,j}$.
• Figure 2: Unsteady vortex from \ref{['sec:vortex']}: convergence test. Left panel: depiction of the initial condition. Right panel: error lines, showing that the scheme is indeed of fifth order accuracy.
• Figure 3: Wet lake at rest from \ref{['sec:lake_at_rest']}: convergence test. Left panel: non-WB scheme; we observe fifth order accuracy. Right panel: WB scheme; we observe an accuracy up to machine precision for each mesh size.
• Figure 4: Wet-dry lake at rest from \ref{['sec:lake_at_rest']}: depiction of the water height (left panel) and of the bathymetry (right panel).
• Figure 5: Wet-dry lake at rest from \ref{['sec:lake_at_rest']}: convergence test. Left panel: non-WB scheme; we observe second order accuracy. Right panel: WB scheme; we observe an accuracy up to machine precision for each mesh size.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3