In this paper we investigate a new class of implicit–explicit (IMEX) two-step methods of Peer type for systems of ordinary differential equations with both non-stiff and stiff parts included in the source term. An extrapolation approach based on already computed stage values is applied to construct IMEX methods with favourable stability properties. Optimised IMEX-Peer methods of order p=2,3,4, 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 implicit–explicit methods are included.
Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
(2017)
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 super-
convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of ex-
plicit 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 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.