High Order Mimetic Symplectic Methods For Hamiltonian Systems

TL;DR

This paper explores high order mimetic spatial schemes combined with symplectic time integrators to accurately preserve energy in Hamiltonian systems, demonstrating improved numerical conservation properties.

Contribution

It introduces high order mimetic spatial operators that preserve key properties of vector calculus, coupled with symplectic schemes for better energy conservation in Hamiltonian simulations.

Findings

High order mimetic operators effectively mimic continuum properties.

Symplectic schemes preserve Hamiltonian energy over time.

Numerical examples confirm improved energy conservation.

Abstract

Hamiltonian systems are known to conserve the Hamiltonian function, which describes the energy evolution over time. Obtaining a numerical spatio-temporal scheme that accurately preserves the discretized Hamiltonian function is often a challenge. In this paper, the use of high order mimetic spatial schemes is investigated for the numerical solution of Hamiltonian equations. The mimetic operators are based on developing high order discrete analogs of the vector calculus quantities divergence and gradient. The resulting high order operators preserve the properties of their continuum ones, and are therefore said to mimic properties of conservation laws and symmetries. Symplectic fourth order schemes are implemented in this paper for the time integration of Hamiltonian systems. A theoretical framework for the energy preserving nature of the resulting schemes is also presented, followed by numerical examples.