An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Abstract

In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.

References in corpus (10)

• High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
• A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
• Well-balanced finite volume evolution Galerkin methods for the shallow water equations
• An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity
• Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes
• Shallow water equations: Split-form, entropy stable, well-balanced, and positivity preserving numerical methods
• DeC and ADER: Similarities, Differences and a Unified Framework
• Asymptotic preserving Deferred Correction Residual Distribution schemes
• Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes
• Issues with Positivity-Preserving Patankar-type Schemes

Cited by in corpus (7)

• Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation
• Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order
• A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity
• On improving the efficiency of ADER methods
• A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations
• Structure-Preserving Numerical Methods for Fokker-Planck Equations
• The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws