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 (2)

• Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes
• Issues with Positivity-Preserving Patankar-type Schemes

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• Structure-Preserving Numerical Methods for Fokker-Planck Equations
• The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws