[Paper Review] A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations

Christophe Chalons, Pierre Kestener+2 more|arXiv (Cornell University)|Apr 6, 2016

Computational Fluid Dynamics and AerodynamicsEngineering31 references22 citations

TL;DR

This paper presents a large time-step, well-balanced Lagrange-Projection scheme for the shallow water equations using an implicit-explicit (IMEX) approach that treats acoustic waves implicitly and material waves explicitly. The method removes the CFL restriction tied to sound speed, enables stable large time steps, and preserves the lake-at-rest steady state while maintaining entropy stability and positivity of water height.

ABSTRACT

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context recent implicit-explicit schemes developed in the framework of compressibleflows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases.

Motivation & Objective

• To develop a numerical scheme for the shallow water equations that overcomes the restrictive CFL condition based on acoustic waves in traditional Godunov-type schemes.
• To preserve the well-balanced property for the lake-at-rest steady state, ensuring exact preservation of still water solutions.
• To maintain positivity of water depth and satisfy a discrete entropy inequality for robustness in subsonic and low-Froude number flows.
• To extend recent IMEX Lagrange-Projection strategies from compressible flows to shallow water systems, enabling large time steps without sacrificing accuracy or stability.
• To validate the scheme on benchmark test cases including fluvial, transcritical, and non-unique Riemann solutions.

Proposed method

• The scheme employs a Lagrange-Projection decomposition to split the system into acoustic and material wave components.
• Acoustic waves are treated implicitly using a semi-implicit time discretization, allowing time steps independent of sound speed.
• Material waves are treated explicitly, preserving accuracy in slow-moving, subsonic regimes.
• The method uses a well-balanced reconstruction to exactly capture the lake-at-rest state, even on non-uniform meshes.
• A conservative entropy formulation is enforced in the discrete scheme to ensure entropy stability.
• The scheme is implemented in a finite-volume framework with numerical fluxes derived from the Lagrange-Projection splitting.

Experimental results

Research questions

• RQ1 Can an IMEX Lagrange-Projection scheme achieve large time steps in shallow water simulations without violating stability or accuracy?
• RQ2 Does the proposed scheme preserve the lake-at-rest steady state (well-balanced property) even with topography and non-uniform meshes?
• RQ3 How does the scheme perform in capturing non-unique entropy solutions to the Riemann problem compared to other methods?
• RQ4 What is the impact of implicit treatment of acoustic waves on the CFL condition and computational efficiency?
• RQ5 Can the scheme maintain positivity of water depth and satisfy a discrete entropy inequality in complex flow regimes?

Key findings

• The scheme achieves stable simulations with time steps up to ten times larger than the explicit CFL limit, as demonstrated in the fluvial regime test case.
• The Lagrange-Projection scheme produces solutions very close to those of the HRHLL scheme with hydrostatic reconstruction, and outperforms the ACU scheme in accuracy.
• In the transcritical regime with shock, the scheme captures the correct shock location and wave structure, matching results from established schemes.
• For the non-unique Riemann problem, the scheme converges to the same solution as the hydrostatic reconstruction method, while HLLACU produces a different, less accurate solution.
• The implicit treatment of acoustic waves removes the sound-speed-based CFL restriction, enabling large time steps while preserving well-balancedness and positivity.
• The scheme maintains entropy stability and positivity of water depth across all test cases, including under strong topography and near-still conditions.