Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration

TL;DR

This paper introduces a symmetric integration method based on Lanczos-Dyche that preserves energy and symplectic structure for linear systems, offering unconditionally stable evolution for PDEs, and shares similarities with Euler-Maclaurin but with superconvergence.

Contribution

It demonstrates that Lanczos-Dyche integration preserves both energy and symplecticity for linear systems, providing a superconvergent, unconditionally stable alternative to traditional methods.

Findings

Preserves energy and symplectic structure for linear systems.

Unconditionally stable for PDE evolution.

Superconvergent compared to Euler-Maclaurin.

Abstract

Numerical evolution of time-dependent differential equations via explicit Runge-Kutta or Taylor methods typically fails to preserve symmetries of a system. It is known that there exists no numerical integration method that in general preserves both the energy and the symplectic structure of a Hamiltonian system. One is thus normally forced to make a choice. Nevertheless, a symmetric integration formula, obtained by Lanczos-Dyche via two-point Taylor expansion (or Hermite interpolation), is shown here to preserve both energy as well as symplectic structure for linear systems. This formula shares similarities with the Euler-Maclaurin formula, but is superconvergent rather than asymptotically convergent. For partial differential equations, the resulting evolution methods are unconditionally stable, i.e, not subject to a Courant-Friedrichs-Lewy limit. Although generally implicit, these methods become explicit for linear systems.