On Adaptive Patankar Runge–Kutta methods

Abstract

We apply Patankar Runge-Kutta methods to y ′ = M (y)y and focus on the case where M (y) is a graph Laplacian as the resulting scheme will preserve positivity and total mass. The second order Patankar Heun method is tested using four test problems (stiff and non-stiff) cast into this form. The local error is estimated and the step size is chosen adaptively. Concerning accuracy and efficiency, the results are comparable to those obtained with a traditional L-stable, second order Rosenbrock method.

Recent Developments in the Field of Modified Patankar‐Runge‐Kutta‐methods

“…However, MPRK methods have not yet been improved with respect to their efficiency. Up to now, only standard step size controllers were used for a particular PRK method [45]. However, the construction of a tailored time step controller is still missing.…”