Positivity-preserving schemes of higher order do not belong to the class of general linear methods. As an example of such higher order schemes, we present Patankar-type methods, which are also conservative. However, an analytic investigation of these schemes is more complex due to their nonlinear nature. 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 test problem. To illustrate the theoretical results, we consider several members of the modified Patankar-type family within numerical experiments.
Thomas is a PhD student in Numerical Analysis at the University of Kassel, Germany. His PhD thesis work deals with a class of numerical methods for solving initial value problems. More precisely, among Thomas' research interests are Modified Patankar-Runge-Kutta schemes and their stability, their dense-output formulas and time step control.