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

Abstract

In this paper, we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203-215, 2017] to a broader class of two-step methods that allow the construction of superconvergent IMEX-Peer methods with A-stable 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 superconvergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included.

Explicit and implicit error inhibiting schemes with post-processing

“…Constructing IMEX-EIS+ methods. Schneider, Lang, and Hundsdorfer [23] showed how to construct IMEX methods of the form (4) which satisfy the order conditions to order p but are error inhibiting and so produce a solution of order p + 1. In this section we show that under additional conditions we can express the exact form of the next term in error and define an associated post-processor that allow us to recover order p + 2 from a scheme that would otherwise be only pth-order accurate.…”

“…In [23] it was shown that if the truncation error vector τ n p+1 lives in the null-space of the operator D then the order of the error is of order p+1. In the following theorem we establish additional conditions on the coefficient matrices D, A F , R F , A G , R G , which allow us to determine precisely what the leading term of this error will look like and therefore remove it by 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.…”

“…The comparison is limited by several factors: first, the methods in [23] are not only diagonally implicit, but singly diagonally implicit (i.e. the lower triangular matrix R G has a diagonal that is all the same number), which makes them different from ours; second, we were not able to find an IMEX-EIS+(2,3) method, so we compare our IMEX-EIS(2,3) to the corresponding (but singly diagonally implicit) method in [23]; finally, the (4,5) method in [23] was obtained by loosening the conditions on the matrix D and requiring only that D k τ n 5 = 0 for some value k, so this does not exactly satisfy the condition (19a), but still gives a fifth order results. Subject to all these caveats in the comparison of the methods, we consider the stability regions in Figure 1 and observe that our methods have larger imaginary axis stability for the same value of s, P .…”

“…1. Linear stability regions for the explicit case G = 0 for our methods compared to methods in [23]. Left: IMEX-EIS(2,3) method (blue) compared to the IMEX-Peer2s, s=2 optimally zerostable method (red) in [23].…”