Patankar-Type Runge-Kutta Schemes for Linear PDEs
For researchers in numerical PDEs, this work provides an incremental improvement to Patankar-type schemes for handling wetting-drying conditions in shallow-water flows.
The paper analyzes the local discretization error of Patankar-type Runge-Kutta methods for linear PDEs, showing second-order accuracy for a known scheme but large errors in drying regions. A modified scheme with an explicit testing stage improves accuracy in wetting-drying scenarios.
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 ${\cal O}(Δt^3)$ 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.