Functional-preserving predictor-corrector multiderivative schemes

TL;DR

This paper introduces high-order multiderivative time integration methods that preserve functionals discretely, utilizing Hermite-Birkhoff-Predictor-Corrector techniques and relaxation, resulting in improved schemes with minimal additional cost.

Contribution

It presents a novel class of functional-preserving multiderivative schemes combining predictor-corrector methods with relaxation techniques.

Findings

Relaxed methods outperform unrelaxed ones.

High-order schemes achieve better accuracy.

Minimal computational overhead for relaxation.

Abstract

In this work, we develop a class of high-order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite-Birkhoff-Predictor-Corrector methods and the technique of relaxation for numerical methods of ODEs. We explain the algorithm in detail and show numerical results for two- and three-derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained.