Symplectic GARK methods for partitioned Hamiltonian systems

TL;DR

This paper introduces symplectic GARK methods tailored for partitioned Hamiltonian systems, enabling efficient, structure-preserving numerical solutions that handle multiple energy components and time scales.

Contribution

It develops symplectic GARK schemes with conditions for symplecticness, symmetry, and time-reversibility, specifically for partitioned Hamiltonian systems with multiple energy components.

Findings

Derived conditions for symplecticness, symmetry, and reversibility.

Constructed schemes for multi-energy Hamiltonian systems.

Demonstrated efficiency in handling different time scales.

Abstract

Generalized Additive Runge-Kutta schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work develops symplectic GARK schemes for additively partitioned Hamiltonian systems. In a general setting, we derive conditions for symplecticness, as well as symmetry and time-reversibility. We show how symplectic and symmetric schemes can be constructed based on schemes which are only symplectic, or only symmetric. Special attention is given to the special case of partitioned schemes for Hamiltonians split into multiple potential and kinetic energies. Finally we show how symplectic GARK schemes can leverage different time scales and evaluation costs for different potentials, and provide efficient numerical solutions by using different order for these parts.