Super-convergent implicit–explicit Peer methods with variable step sizes

Abstract

Dynamical systems with sub-processes evolving on many different time scales are ubiquitous in applications. Their efficient solution is greatly enhanced by automatic time step variation. This paper is concerned with the theory, construction and application of IMEX-Peer methods that are super-convergent for variable step sizes and A-stable in the implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods which keep their higher order for variable step sizes and exhibit favourable linear stability properties, we derive necessary and sufficient conditions on the nodes and coefficient matrices and apply an extrapolation approach based on already computed stage values. New super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are given as result of additional order conditions which maintain the super-convergence property independent of step size changes. Numerical experiments and a comparison to other super-convergent IMEX-Peer methods show the potential of the new methods when applied with local error control.

Alternating directions implicit integration in a general linear method framework

“…The theory of partitioned GLMs was formalized in [20] and different families of methods based on this structure have been reported in [21][22][23][24]. More recent high order IMEX-GLMs found in the literature [25][26][27] are based on Diagonally Implicit Multistage Integration Methods (DIMSIMs), Two-Step Runge-Kutta methods, and Peer methods providing various accuracy and stability enhancements.…”

Well-Balanced and Asymptotic Preserving IMEX-Peer Methods

“…There is a wide range of literature concerning the different aspects of Peer methods and we will only give a short overview. More details can be found in the introductory chapters of [9,10]. Peer methods were introduced by Schmitt and Weiner in 2004 [8].…”

“…An alternative construction using partitioned methods is given in [11,16]. Since the coefficient matrices of Peer methods offer many degrees of freedom, the construction of super-convergent schemes [9,13,15] and the adaptation to variable step sizes [10,12] is possible.…”

“…For the sake of notation, we use for an s × s matrix M the same symbol for its Kronecker product with the m × m identity matrix M ⊗ I m as a mapping from the space R s•m to itself. An extensive analysis of consistency and stability along with the construction of super-convergent methods as well as the adaption to variable step sizes is given in [9,10].…”

Explicit and implicit error inhibiting schemes with post-processing

“…In recent work, Schneider and colleagues [23,24] produced super-convergent IMEX Peer methods (a Peer method is a GLM where each stage is of the same order). These methods satisfy error inhibiting conditions similar to those in [5] and so produce order p + 1 although their truncation error is of order p, so we shall refer to them here as IMEX-EIS schemes.…”