A new stable splitting for singularly perturbed ODEs

Abstract

In this publication, we consider IMEX methods applied to singularly perturbed ordinary differential equations. We introduce a new splitting into stiff and non-stiff parts that has a direct extension to systems of conservation laws and investigate its performance analytically and numerically. We show that this splitting can in some cases improve the order of convergence, demonstrating that the phenomenon of order reduction is not only a consequence of the method but also of the splitting.

A New Stable Splitting for the Isentropic Euler Equations

“…As mentioned before we define the splitting into stiff and non-stiff parts by linearization around the reference solution w (0) . The splitting idea has been analyzed in the context of ODEs in [41], a first extension to isentropic Euler equations was given in [42]. Similar ideas have been used before in [23].…”

“…In [41], we investigated the performance of the RS-IMEX splitting in the setting of singular perturbed ODEs and were able to obtain improved stability and accuracy results in comparison to standard splittings. This earlier work serves as a motivation to use the RS-IMEX also for the isentropic Euler equations.…”

“…To overcome this difficulty, we propose a general splitting, which is linear in the implicit part, based on a reference solution and call this splitting RS-IMEX [41]. In the present paper, we choose the zeroth-order asymptotic component w (0) as reference solution.…”

“…The RS-IMEX approach was first analyzed in [41] for a singularly perturbed system of ODEs, and gave superior results for high order IMEX Runge-Kutta and BDF time discretizations. Furthermore, in [47] Zakerzadeh et al proved stability of the modified equation for the one-dimensional shallow water equations with flat bottom.…”

“…This concept is not restricted to low Mach flows, but it can also be applied to other discretizations of singular problems. Examples are the discretization of kinetic equations, see, e.g., [11,29,34,37] and the references therein, or the discretization of singularly perturbed ODEs, see, e.g., [9,10,25,41] and the references therein. This list is by no means exhaustive, we refer to [28] for a more detailed overview.…”