Sigrun Ortleb

Sigrun Ortleb, Willem Hundsdorfer

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.

Anita Gjesteland, Sigrun Ortleb, Salim Elghawi et al.

We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer formulated as a hyperbolic balance law in diffusive scaling for a particle distribution function of the independent variables space, time and velocity. Our numerical discretization is based on the well-known technique of micro-macro decomposition which results in a system of balance laws for equilibrium and non-equilibrium quantities and facilitates preservation of the asymptotic limit for vanishing scaling parameters at the discrete level. We prove fully discrete stability and asymptotic preservation for general spatial and temporal discretizations having the summation-by-parts property. A new provably energy-stable Dirichlet boundary treatment for the micro-macro decomposed system is developed based on the introduction of simultaneous approximation terms. Numerical results show convergence for smooth problems and demonstrate energy stability of the proposed boundary treatment.