Explicit vs implicit schemes for the spectral method for the heat equation

Abstract

Three different boundary condition enforcement/time-stepping formulations are detailed for the Chebyshev collocation spectral method. Application is made to the one-dimensional heat equation with a saw-tooth initial condition and several boundary condition combinations. It is observed that for a small time step, the explicit time stepping/strong boundary condition implementation method and the implicit time-stepping method give essentially the same solutions for various numbers of collocation points. The explicit time-stepping/weak boundary condition enforcement scheme has significantly larger error at low numbers of collocation points, and requires considerably more collocation points to achieve convergence. While implicit schemes can employ significantly larger time steps than can the explicit schemes, implicit schemes require some form of matrix inversion, and this can be costly. Conversely, the explicit schemes can exploit the use of fast transform techniques to economize each time-step solution, but stability considerations limit the size of the time step that can be employed. A series of parametric studies are conducted to assess the relative economies of explicit vs implicit time-stepping schemes. The one-dimensional heat equation is used as the basis problem and Robin boundary conditions are considered. The maximum explicit time-step limit is determined and a correlation provided that includes the dependence on boundary condition parameter values.