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

## TL;DR

This paper develops super-convergent IMEX-Peer methods that maintain high order accuracy with variable step sizes and good stability, enhancing the efficient numerical solution of multi-scale dynamical systems.

## Contribution

The paper introduces new super-convergent IMEX-Peer methods of orders 3, 4, and 5, with conditions ensuring high order accuracy under variable step sizes.

## Key findings

- New IMEX-Peer methods of order s+1 for s=2,3,4 stages.
- Methods maintain super-convergence with variable step sizes.
- Numerical experiments demonstrate improved performance.

## 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.