Symplectic finite-difference methods for solving partial differential equations

TL;DR

This paper introduces symplectic finite-difference methods for solving PDEs, which improve stability and qualitative accuracy by leveraging symplectic integrator principles, enabling high-order, stable algorithms for diffusion and advection equations.

Contribution

It establishes a systematic connection between symplectic integrators and finite-difference schemes, allowing the development of high-order, stable algorithms for PDEs.

Findings

Symplectic diffusing algorithm is unconditionally stable.

Symplectic advection algorithm is unitary.

High-order algorithms can be systematically constructed.

Abstract

The usual explicit finite-difference method of solving partial differential equations is limited in stability because it approximates the exact amplification factor by power-series. By adapting the same exponential-splitting method of deriving symplectic integrators, explicit symplectic finite-difference methods produce Saul'yev-type schemes which approximate the exact amplification factor by rational-functions. As with conventional symplectic integrators, these symplectic finite-difference algorithms preserve important qualitative features of the exact solution. Thus the symplectic diffusing algorithm is {\it unconditionally stable} and the symplectic advection algorithm is {\it unitary}. There is a one-to-one correspondence between symplectic integrators and symplectic finite-difference methods, including the key idea that one can systematically improve an algorithm by matching its modified Hamiltonian more closely to the original Hamiltonian. Consequently, the entire arsenal of symplectic integrators can be used to produce arbitrary high order time-marching algorithms for solving the diffusion and the advection equation.