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arXiv · 1610.02715

Patankar-Type Runge-Kutta Schemes for Linear PDEs

Sigrun Ortleb·Willem Hundsdorfer
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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 ${\cal O}(\Delta t^3)$ 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.

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Sigrun Ortleb, Willem Hundsdorfer. 2016-10-09. Patankar-Type Runge-Kutta Schemes for Linear PDEs. https://doi.org/10.1063/1.4992489

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