Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.
翻译：修改后的Patankar-Runge-Kutta (MPRK) 方法保存了生产-销毁系统(PDS)的所有时间级大小的相对性和保守性,因此,Ptankar-Runge-Kutta (MPRK) 方法不属于一般线性方法的类别, 即即使PDS是线性的, 也由非线性地图 $\mathbf g$ 生成。 此外, 由于该方法的保守性, $\mathbf g$ 地图拥有非Hyperblic 固定点的固定点。 最近, 开发了用于调查非线性迭代图非Hyperblic 固定点稳定性特性的新理论。 我们应用该理论来理解对普通差异方程式的非线性 PRTRK 方法组的稳定性特性。 显示, 固定点对于所有时间级大小和MPRK 家庭成员来说都是稳定的。 最后, 实验用数字支持理论性索赔 。
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