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

Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes