A comprehensive theory on generalized BBKS schemes

Abstract

In the past years several unconditionally positivity preserving numerical methods for production-destruction equations arising in a wide variety of real life applications were developed. The BBKS schemes suggested by Bruggemann et al. (2007) [4] and Broekhuizen et al. (2008) [3] modifications of Heun's method and conserve all linear invariants while still maintaining positivity independent of the time step size used. In this paper, we present a comprehensive generalization of these schemes as modifications of arbitrary first and second order Runge-Kutta methods with nonnegative parameters. This is achieved by the introduction of novel Patankar weights and sufficient and necessary conditions w.r.t. the Patankar weight denominators for first as well as second order consistency are proven. Furthermore, a convergence proof for the incorporated Newton scheme is given such that the efficiency compared to the original BBKS schemes, which make use of the bisection method is significantly improved. Numerical simulations are presented confirming the theoretical results concerning the validity, efficiency and accuracy of the new class of generalized BBKS schemes.