Comparison of the asymptotic stability properties for two multirate strategies

Abstract

This paper contains a comparison of the asymptotic stability properties for two multirate strategies. For each strategy, the asymptotic stability regions are presented for a 2 ×2 test problem and the differences between the results are discussed. The considered multirate schemes use Rosenbrock type methods as the main time integration method and have one level of temporal local refinement. Some remarks on the relevance of the results for 2 × 2 test problems are presented.

Multirate Numerical Integration for Stiff ODEs

“…In this paper we made on overview and extended the multirate time stepping strategy introduced in [7,10,11,12]. As seen from the numerical tests, the efficiency of time integration methods can be significantly improved by using large time steps for inactive components, without sacrificing accuracy.…”

“…A multirate method based on the Rosenbrock methods, together with a self-adjusting partitioning strategy was introduced and analyzed in [7,10,11,12]. In this paper we present an overview of this method and suggest a way to improve it.…”

Construction of a multirate RODAS method for stiff ODEs

“…An interpolant with this property was considered together with a second-order Rosenbrock method in [11]. This combination resulted in a multirate method which showed good asymptotic stability properties.…”

Multirate generalized additive Runge Kutta methods

“…The use of dense output interpolation for coupling the slow and fast components was developed by Savcenco, Hundsdorfer, and co-workers in the context of Rosenbrock methods [12][13][14][15][16]. This approach can be immediately extended to Runge Kutta methods, and the overall scheme can be formulated in the mutirate GARK framework.…”

“…However, the stability conditions (13) cannot be fulfilled when the simplifying conditions (16a, 16b) hold.Theorem 3 (Internally consistent multirate GARK schemes are not stability decoupled) When only the first fast microstep is coupled to the slow part (10), the stability conditions(13) are not compatible with the first simplifying condition (16a) for multirate GARK schemes with M > 1.…”