On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters

Abstract

AbstractRecently, a stability theory has been developed to study the linear stability of modified Patankar–Runge–Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge–Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($$\varvec{\alpha }$$ α ) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($$\alpha $$ α ) schemes with nonnegative Runge–Kutta parameters. In particular, it is shown that MPRK22($$\alpha $$ α ) schemes with $$0<\alpha <0.5$$ 0 < α < 0.5 or $$-0.5<\alpha <0$$ - 0.5 < α < 0 are only stable if the time step size is sufficiently small. But schemes with $$\alpha \le -0.5$$ α ≤ - 0.5 are stable and converge towards the steady state of the initial value problems for all time step sizes, at least for the test problem under consideration. Furthermore, it is shown that for some of the latter systems, the initial values must actually be close enough to the steady state to guarantee this result.