On Lyapunov stability of positive and conservative time integrators and application to second order modified Patankar–Runge–Kutta schemes

Abstract

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.

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

“…Furthermore, (11b) then follows from Lemma 19. Finally, computing y n+1 according to (3), also taking into account equation (10), we obtain…”

“…Corollary 5. Let u be defined as in (10) and the stage equations of the NSARK method possess a solution for small enough h. Furthermore, let A [ν] (y n , h) = O(1) and F [ν] ∈ C p+1 for ν = 1, . .…”

“…Up to now, only conditions for first and second order GeCo schemes are available. Furthermore, due to the complex structure of the functions ϕ i and ϕ n+1 from [16], a stability analysis was not available until recently [10].…”

“…Modified Patankar-Runge-Kutta methods (MPRK) are applied to so-called production-destruction systems (PDS) m for all n ∈ N. Besides these properties, MPRK schemes demonstrated their robustness many times by solving also stiff problems [3,14,15]. Recently, a Lyapunov stability analysis has been published underlying their robustness observed in numerical experiments [11,12,13].…”

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.…”