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.

翻译：近年来，Patankar格式因其能够在任意选取的时间步长下保持生产-破坏系统（PDS）解析解的正性而受到越来越多的关注。尽管这些格式目前备受关注，但长期以来其稳定性性质尚不明确。最近，基于李雅普诺夫稳定性并扩展中心流形定理的一种新的稳定性方法被提出，用于研究保正时间积分器的稳定性性质。本文研究了经典修正的Patankar--Runge--Kutta格式（MPRK）和修正的Patankar延迟校正（MPDeC）方法的稳定性性质。我们证明了大多数所考虑的MPRK格式对任意时间步长都是稳定的，并计算了MPDeC的稳定性函数。通过数值分析，我们进一步研究了其性质，发现大多数MPDeC格式同样对任意选取的时间步长保持稳定。最后，我们通过数值模拟验证了理论结果。