A Time-Spectral Method for Initial-Value Problems Using a Novel Spatial Subdomain Scheme

Abstract

We analyse a new subdomain scheme for a time-spectral method for solving initial boundary value problems. Whilst spectral methods are commonplace for spatially dependent systems, finite difference schemes are typically applied for the temporal domain. The Generalized Weighted Residual Method (GWRM) is a fully spectral method in that it spectrally decomposes all specified domains, including the temporal domain, with multivariate Chebyshev polynomials. The Common Boundary-Condition method (CBC) is a spatial subdomain scheme that solves the physical equations independently from the global connection of subdomains. It is here evaluated against two finite difference methods. For the linearised Burger equation the CBC-GWRM is $\sim30\%$ faster and $\sim50\%$ more memory efficient than the semi implicit Crank-Nicolson method at a maximum error $\sim10^{-5}$. For a forced wave equation the CBC-GWRM manages to average efficiently over the small time-scale in the entire temporal domain. The CBC-GWRM is also applied to the linearised ideal magnetohydrodynamic (MHD) equations for a screw pinch equilibrium. The growth rate of the most unstable mode was efficiently computed with an error $<0.1\%$.