A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods

Abstract

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.

Recent Developments in the Field of Modified Patankar‐Runge‐Kutta‐methods

“…Moreover, in the case of stable fixed points, the iterates of the MPRK method are proved to locally converge towards the correct steady state solution. The stability properties as well as the local convergence could be observed in numerical experiments [39,41]. Indeed, it turned out that MPRK22(α) and MPRK43(γ) are stable in this sense for all parameter choices [39,41].…”

“…However, as MPRK schemes do not belong to the class of general linear methods, a new approach was used in [38,39] to investigate their stability properties. The resulting theory is based on the center manifold theorem for maps [49,50] and was applied to second-order MPRK22(α) [39] and to thirdorder MPRK methods MPRK43(α, β) and MPRK43(γ) in [41]. To that end, the linear problem y ′ (t) = Λy(t), y(0…”

“…Due to the nonlinear nature of these Patankar-type methods, a stability analysis for several members of this family of schemes was not available until recently [38][39][40][41][42], where the local behavior near steady states of linear autonomous problems was investigated using the theory of discrete dynamical systems. We also want to note that some issues of Patankar-type schemes were discussed in [43], where different modified Patankar methods from [29,30,33] were analyzed with respect to oscillatory behavior.…”

An arbitrary high order and positivity preserving method for the shallow water equations

“…There are several perspectives to this work. They and range from deep questions on the numerical analysis and stability of modified Patankar schemes, which is an open research topic [63,32,33], especially when coupled to space discretizations in the context of PDEs, to the possible development of this approach on unstructured meshes to exploit advanced mesh adaptation algorithms to capture the flow features with even better resolution, and to save even more computational resources.…”

“…Both DeC methods and Patankar trick have a long history. In particular, for more information on DeC the interested reader is referred to [22,44,45,64,28], while Patankar (and modified Patankar) tricks are detailed in [53,51,13,14,33,35,36,30,31,41,52].…”