A time-spectral method for initial-value problems using a novel spatial subdomain scheme. Issue 1 (1st January 2018) - Lindvall, Kristoffer; Scheffel, Jan
Abstract: We analyse a novel subdomain scheme for time-spectral solution of initial-value partial differential equations. In numerical modelling spectral methods are commonplace for spatially dependent systems, whereas finite difference schemes are typically applied for the temporal domain. The Generalized Weighted Residual Method (GWRM) is a fully spectral method that spectrally decomposes all specified domains, including the temporal domain, using multivariate Chebyshev polynomials. The Common Boundary-Condition method (CBC) here proposed is a spatial subdomain scheme for the GWRM. It solves the physical equations independently from the global connection of subdomains in order to reduce the total number of modes. Thus, it is a condensation procedure in the spatial domain that allows for a simultaneous global temporal solution. It is here evaluated against the finite difference methods of Crank-Nicolson and Lax-Wendroff for two example linear PDEs. 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 % .