Recently, 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($\alpha$) 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($\alpha$) schemes with nonnegative Runge-Kutta parameters.

翻译：最近，针对修正Patankar–Runge–Kutta（MPRK）格式的线性稳定性，已发展出相应的稳定性理论。该稳定性理论给出了MPRK格式不动点稳定性的充分条件，以及MPRK格式收敛至相应初值问题稳态的充分条件，其主要假设是初值充分接近稳态状态。最初，多项出版物中的数值实验表明，这些线性稳定性性质不仅是局部的，而且是全局的——这与一般线性方法的情况类似。然而，近期发现MPDeC(8)格式的线性稳定性本质上仅具有局部性质。我们推测，这是由于MPDeC(8)格式的负Runge–Kutta（RK）参数所致；若RK参数非负，则线性稳定性确实为全局性质。为支持这一猜想，我们考察了具有负RK参数的MPRK22($\alpha$)方法族，并表明即使在这些方法中，也存在稳定性性质仅为局部的情形。然而，对于具有非负Runge–Kutta参数的MPRK22($\alpha$)格式，并未观察到这种局部线性稳定性。