Stability of Spectral Collocation Schemes with Explicit-Implicit-Null Time-Marching for Convection-Diffusion and Convection-Dispersion Equations

In this paper, we discuss the Fourier collocation and Chebyshev collocation schemes coupled with two specific high order explicit-implicit-null (EIN) time-marching methods for solving the convection-diffusion and convection-dispersion equations. The basic idea of the EIN method discussed in this paper is to add and subtract an appropriate large linear highest derivative term on one side of the considered equation, and then apply the implicit-explicit time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. We give stability analysis for the proposed EIN Fourier collocation schemes on simplified linear equations by the aid of the Fourier method. We show rigorously that the resulting schemes are stable with particular emphasis on the use of large time steps if appropriate stabilization parameters are chosen. Even though the analysis is only performed on the EIN Fourier collocation schemes, numerical results show that the stability criteria can also be extended to the EIN Chebyshev collocation schemes. Numerical experiments are given to demonstrate the stability, accuracy and performance of the EIN schemes for both one-dimensional and two-dimensional linear and nonlinear equations.