论文标题
关于正面和保守的时间集成商的lyapunov稳定性以及对二阶修改的patankar-运行的应用 - kutta方案
On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes
论文作者
论文摘要
从近二十年来开始，经过修改的patankar-runge-kutta（MPRK）方法已被证明是有效且健壮的数值方案，可在所选择的时间步长的时间步长中保留生产摧毁系统的积极性和保守性。由于这些有利的特性，它们用于多种应用。然而，直到现在，仍然缺少对MPRK方案稳定性的分析研究，因为通常通过Dahlquist方程的方法是不可行的。因此，我们考虑了一个积极而保守的2D测试问题，并提供了基于中心歧管理论对一般正面和保守的时间整合器方案的稳定性分析可用的陈述。我们使用这种方法来研究第二阶MPRK22（$α$）和MPRK22NC（$α$）方案的Lyapunov稳定性。我们证明MPRK22（$α$）方案无条件稳定，并得出MPRK22NC（$α$）方案的稳定区域。最后，提出了数值实验，以证实理论结果。
Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Nevertheless, until now, an analytic investigation of the stability of MPRK schemes is still missing, since the usual approach by means of Dahlquist's equation is not feasible. Therefore, we consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive and conservative time integrator schemes based on the center manifold theory. We use this approach to investigate the Lyapunov stability of the second order MPRK22($α$) and MPRK22ncs($α$) schemes. We prove that MPRK22($α$) schemes are unconditionally stable and derive the stability regions of MPRK22ncs($α$) schemes. Finally, numerical experiments are presented, which confirm the theoretical results.
