@article{SchneiderLangHundsdorfer, author = {Schneider, Moritz and Lang, Jens and Hundsdorfer, Willem}, title = {Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part}, volume = {J. Comput. Physics}, number = {Vol. 367}, doi = {10.1016/j.jcp.2018.04.006}, pages = {121 -- 133}, 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 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.}, language = {en} } @article{SchneiderLangWeiner2019, author = {Schneider, Moritz and Lang, Jens and Weiner, R{\"u}diger}, title = {Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes}, volume = {J. Comput. Appl. Math.}, number = {387}, doi = {doi:10.1016/j.cam.2019.112501}, pages = {112501}, year = {2019}, 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.}, language = {en} }