Structural schemes for hamiltonian systems

TL;DR

This paper adapts the structural method for Hamiltonian systems, enabling stable, high-order accurate discretizations that preserve invariants, and extends it from scalar to vector and non-separable systems with demonstrated efficiency.

Contribution

The paper introduces a specialized structural scheme for Hamiltonian systems, extending the method to vector and non-separable cases, with a focus on stability and invariant preservation.

Findings

The scheme achieves unconditional stability.

It preserves invariant quantities like total energy.

Numerical results show high efficiency and accuracy.

Abstract

We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem into two sets of equations: the physical equations, which describe the local dynamics of the system, and the structural equations, which only involve the discretization on a very compact stencil. They have desirable properties, such as unconditional stability or high-order accuracy. We first give a general description of the scheme for the scalar case (which corresponds to e.g. spring-mass interactions or pendulum motion), before extending the technique to the vector case (treating e.g. the $n$-body system). The scheme is also written in the case of a non-separable system (e.g. a charged particle in an electromagnetic field). We give numerical evidence of the method's efficiency, its capacity to preserve invariant quantities such as the total energy, and draw comparisons with the traditional symplectic methods.