Applied Mathematics Colloquium: Andreas Meister (U. Kassel)

Title: Higher Order Positivity Preserving Methods for Advection-Diffusion-Reaction Equations

Abstract: Modified Patankar-Runge-Kutta (MPRK) schemes are numerical methods
for the solution of positive and conservative production-destruction systems.
They adapt explicit Runge-Kutta schemes in a way to ensure positivity and
conservation irrespective of the time step size.

We present order conditions for various Patankar-type schemes as well as
a new stability approach that examines the non-hyperbolic fixed points of the
schemes for a general linear test problem. We formulate sufficient conditions for
the stability of such non-hyperbolic fixed points as well as the local convergence
of the numerical approximation towards the correct steady state solution of the
underlying conservative differential equation. To illustrate the theoretical results, 

we consider several members of the modified Patankar-type family within
numerical experiments.

Finally, we compare MPRK schemes with well-known Runge-Kutta methods
in the context of convection-diffusion-reaction equations with source terms of
production-destruction type as well as reactive Euler equations of gas dynamics.