论文标题
对修改的Patankar DEC和高阶改性Patankar RK方法的局部动力学的研究
A study of the local dynamics of modified Patankar DeC and higher order modified Patankar RK methods
论文作者
论文摘要
近年来，Patankar计划在近年来引起了人们的兴趣，因为它们保持生产毁灭系统（PDS）的分析解决方案的阳性，而与所选的时间步长无关。尽管它们现在引起了极大的兴趣，但长期以来，尚不清楚这些方案具有哪些稳定性。最近，已经提出了一种基于Lyapunov稳定性的新稳定方法，并扩展了中心歧管定理，以研究保留阳性时间积分器的稳定性。在这项工作中，我们研究了经典修改的patankar-runge-kutta方案（MPRK）的稳定性和修改后的Patankar延期校正（MPDEC）方法。我们证明，大多数考虑的MPRK方案对于任何时间步长稳定，并计算MPDEC的稳定性函数。我们在数值上研究了它的属性，这些属性也表明大多数MPDEC都稳定，而与所选的时间步长无关。最后，我们通过数值模拟验证了我们的理论结果。
Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity-preserving time integrators. In this work, we study the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.
