论文标题
对非线性生产毁灭系统的改良Patankar-Runge-Kutta方法的稳定性分析
A Stability Analysis of Modified Patankar-Runge-Kutta methods for a nonlinear Production-Destruction System
论文作者
论文摘要
修改的patankar-runge-kutta（MPRK）方法可保留对所有时间步长的普通微分方程的生产摧毁系统（PDS）的阳性和保守性。结果，高阶MPRK方案不属于常规线性方法类别，即，即使PDS是线性的，迭代也是由非线性MAP $ \ MATHBF G $生成的。此外，由于该方法的保守性，地图$ \ mathbf g $具有非纤维固定点。 最近，开发了一种新定理，用于研究非线性迭代图的非纤维固定点的稳定性特性。当将此定理应用于普通微分方程的非线性PD时，我们将此定理用于了解二阶MPRK方法的稳定性。结果表明，固定点对于所有时间步长和MPRK家族的成员都是稳定的。最后，提出实验以在数值上支持理论主张。
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.
