Two novel classes of arbitrary high-order structure-preserving algorithms for canonical Hamiltonian systems

TL;DR

This paper introduces two new classes of high-order algorithms for canonical Hamiltonian systems that preserve structure, including symplecticity and energy invariants, with broad applicability and verified effectiveness.

Contribution

The paper develops two novel classes of structure-preserving algorithms: parameterized symplectic schemes and energy-preserving schemes, extending classical methods with arbitrary order accuracy.

Findings

Schemes are symplectic for any fixed parameter.

Energy-preserving schemes maintain invariants at each step.

Numerical experiments confirm theoretical properties.

Abstract

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.