Relaxation Runge-Kutta methods with first-same-as-last structure

Abstract

Preserving invariants of differential equations under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently, relaxation methods have been proposed as small modifications of standard time integration schemes guaranteeing the correct evolution of functionals of the solution. We investigate how to combine relaxation with error-based step size control for explicit Runge-Kutta methods using the first-same-as-last (FSAL) technique and demonstrate the improved efficiency of our new methods.