D-splitting methods: 2N -storage embedded explicit Runge-Kutta methods at any order using splitting methods

TL;DR

This paper introduces D-splitting methods as high-performance, low-storage explicit Runge-Kutta schemes that preserve qualitative properties of solutions, suitable for efficient PDE integration.

Contribution

It demonstrates that D-splitting methods can serve as 2N-storage embedded explicit RK methods, extending high-order accuracy without additional memory.

Findings

D-splitting methods achieve high order with 2N-storage.

They preserve qualitative properties of solutions.

Numerical tests confirm their effectiveness.

Abstract

Low-storage explicit Runge-Kutta schemes are particularly popular for the numerical integration of time-dependent partial differential equations based on the method-of-lines due to their efficiency and their reduced memory requirements. We show that D-splitting methods, splitting methods on the extended phase space, can be used as high performance 2N-storage embedded explicit RK methods without a third storage register. They are pseudo-geometric methods preserving some of the qualitative properties of the exact solution up to a higher order than the order of the method. Some of their properties are analysed, to build new tailored methods, and are tested on numerical examples.