A Sylvester-Based IMEX Method via Differentiation Matrices for Solving Nonlinear Parabolic Equations (Communications in Computational Physics, 2013)
In this paper we describe a new pseudo-spectral method to solve numerically two- and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomialsas basis functions. The idea is to discretize the equations by means of differentiationmatrices and to relate them to Sylvester-type equations by means of a fourth-orderimplicit-explicit scheme, being of particular interest the treatment of three-dimensionalSylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code.We test numerically the three choices of basis functions, showing the convenience ofthis new approach, especially when rational Chebyshev polynomials are considered.