Space-time spectral collocation methods for the magnetohydrodynamics equations

Abstract

Spectral methods are used to solve partial differential equations numerically. When the solution is analytic, the rate of convergence of the numerical solution is exponential; that is, the error decays exponentially. In time-dependent PDEs, low order finite difference schemes and spectral schemes are traditionally being used for the time and spatial derivatives, respectively. However, applying spectral schemes in both space and time has been thought of recently. These methods have spectral convergence in both spatial and temporal domains. In this thesis, both Chebyshev and Legendre spectral collocation methods are implemented for the Navier–Stokes and Magnetohydrodynamics equations. Numerical solutions for both equations converge exponentially when the solutions are analytic. Moreover, Navier-Stokes and Magnetohydrodynamic equations are implemented for high Reynolds numbers using nonlinear preconditioning methods, ASPIN and RASPEN, which are defined using spectral domain decomposition.