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.