Explicit Exactly Energy-conserving Methods for Hamiltonian Systems

TL;DR

This paper introduces a new explicit energy-conserving numerical method for Hamiltonian systems that is unconditionally stable and computationally efficient, expanding the toolkit for simulating physical systems with energy preservation.

Contribution

It presents a fully explicit, energy-conserving numerical scheme for Hamiltonian systems under non-negativity conditions, avoiding iterative solutions and maintaining stability.

Findings

The method exactly conserves numerical energy for a broad class of Hamiltonian systems.

It is unconditionally stable and computationally comparable to simple integrators like Stormer-Verlet.

Numerical experiments demonstrate effectiveness on classical and PDE systems.

Abstract

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Stormer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam, as well as for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.