Using Implicit ODE Methods with Iterative Linear Equation Solvers in Spectral Methods

Abstract

The use of Chebyshev spectral methods in space on an evolution partial differential equation results, in many cases, in a stiff system of ordinary differential equations (ODEs). ODE solvers based on explicit methods will therefore be inefficient for such equations. The use of conventional implicit ODE solvers is difficult since the Jacobian matrix of the ODE system is full and large.It is shown how to apply ODE solvers with iterative linear equation solvers to ODEs coming from spectral discretizations. These ODE solvers do not use the Jacobian explicitly, but good preconditioners for the Newton matrix are needed for their efficient operation. Preconditioners of the tensor product type are developed for certain classes of hyperbolic partial differential equations (PDEs), and numerical experiments show that these preconditioners give a good performance of the ODE solver and are a substantial improvement over the performance of explicit solvers.