Patankar-Type Runge-Kutta Schemes for Linear PDEs

Abstract

We study the local discretization error of Patankar-type Runge-Kutta methods applied to semi-discrete PDEs. For a known two-stage Patankar-type scheme the local error in PDE sense for linear advection or diffusion is shown to be of the maximal order $\mathcal{O}(\Delta t^3)$ for sufficiently smooth and positive exact solutions. However, in a test case mimicking a wetting-drying situation as in the context of shallow-water flows, this scheme yields large errors in the drying region. A more realistic approximation is obtained by a modification of the Patankar approach incorporating an explicit testing stage into the implicit trapezoidal rule.

5 pages, 4 figures, Submitted to AIP conference proceedings: Proceedings of the 14th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2016), 19-25 Sep 2016, Rhodes, Greece

Cited by in corpus (2)

• Structure-Preserving Numerical Methods for Fokker-Planck Equations
• Unconditionally Positive and Conservative Third Order Modified Patankar-Runge-Kutta Discretizations of Production-Destruction Systems