Extrapolation-based implicit-explicit general linear methods

Abstract

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A-or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings.

Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part

“…, r, of the stability function p(w, z 0 , z 1 ) are inside of the unit circle. In this paper we will be mainly interested in IMEX schemes which are A-stable with respect to the implicit part z 1 ∈ C. To investigate such methods we consider, similarly as in [7,14,28], the sets…”

Extrapolation-based super-convergent implicit-explicit Peer methods with A-stable implicit part

“…with the stability matrix M (z 0 , 0) = (I − z 0 RS 2 ) −1 (P + z 0 Q + z 0 RS 1 ). Efficient numerical algorithms to compute S α and S E are extensively described in [5,16]. Since S α ⊂ S E , the goal is to construct IMEX-Peer methods for which S E is large and S E \S α is as small as possible for angles α that are close to 90 • .…”

“…This idea was first used by Crouzeix [6] with linear multi-step methods of BDF type. Recently, Cardone, Jackiewicz, Sandu and Zhang [5] applied the extrapolation approach to diagonally implicit multistage integration methods and Lang and Hundsdorfer [16] to implicit Peer methods constructed by Beck, Weiner, Podhaisky and Schmitt [1]. IMEX-Peer methods are competitive alternatives to classic IMEX methods for large stiff problems.…”

Order conditions for general linear methods

“…Such high order and stage order methods with some desirable stability properties (large regions of absolute stability for explicit methods, A-, L-, and algebraic stability for implicit methods) were investigated in a series of papers [29,56,30,[16][17][18][19][20][21][22][23][24][25][26][27]33,34,[57][58][59] and the monograph [3]. In the next section we will derive the general order conditions for GLMs (1.5) without restrictions on stage order (except stage preconsistency and stage consistency conditions (3.11) and (3.12)) using the approach proposed by Albrecht [60][61][62][63][64] in the context of RK, composite and linear cyclic methods, and generalized by Jackiewicz and Tracogna [39,40] and Tracogna [43], to TSRK methods for ODEs, by Jackiewicz and Vermiglio [65] to general linear methods with external stages of different orders, and by Garrappa [66] to some classes of Runge-Kutta methods for Volterra integral equations with weakly singular kernels.…”