Determination of Legendre–Gauss–Lobatto Pseudo–spectral Method for One–Dimensional Advection–Diffusion Equation

Abstract

In this chapter, we describe a Legendre pseudo–spectral approach for solving one–dimensional parabolic advection–diffusion equations with constant parameters subject to a given initial and boundary conditions, based on Legendre–Gauss–Lobatto zeros and tensor product formulation. First, we use differentiation matrices and their derivatives with respect to x and t to approximate the unknown function. Second, we solve our issue by converting it to a linear system of equations with unknowns at the collocation locations. Finally, various examples are shown, together with numerical results, to demonstrate the efficacy of the proposed method.