Spectrally accurate space-time solution of Hamiltonian PDEs

TL;DR

This paper discusses spectrally accurate space-time methods for solving Hamiltonian PDEs, focusing on efficient implementations for stiffly-oscillatory problems derived from semi-discretized PDEs.

Contribution

It introduces a novel approach to efficiently solve stiffly-oscillatory Hamiltonian PDEs using spectral methods in time.

Findings

Effective spectral methods for Hamiltonian PDEs

Improved efficiency for stiffly-oscillatory problems

Enhanced accuracy in space-time solutions

Abstract

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.