Symplectic multirate generalized additive Runge-Kutta methods for Hamiltonian systems

TL;DR

This paper develops symplectic multirate GARK methods for efficiently solving additively partitioned Hamiltonian systems with multiple time scales, ensuring stability, symplecticity, and reversibility.

Contribution

It introduces new symplectic multirate GARK schemes with derived order, symplecticity, and reversibility conditions, and demonstrates their effectiveness on complex Hamiltonian problems.

Findings

Schemes achieve high order accuracy

Implicit-explicit partitions improve stability

Effective for Fermi-Pasta-Ulam problem

Abstract

The generalized additive Runge-Kutta (GARK) framework provides a powerful approach for solving additively partitioned ordinary differential equations. This work combines the ideas of symplectic GARK schemes and multirate GARK schemes to efficiently solve additively partitioned Hamiltonian systems with multiple time scales. Order conditions, as well as conditions for symplecticity and time-reversibility, are derived in the general setting of non-separable Hamiltonian systems. Investigations of the special case of separable Hamiltonian systems are also carried out. We show that particular partitions may introduce stability issues, and discuss partitions that enable an implicit-explicit integration leading to improved stability properties. Higher-order symplectic multirate GARK schemes based on advanced composition techniques are discussed. The performance of the schemes is demonstrated by means of the Fermi-Pasta-Ulam problem.