Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Abstract

This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

Numerical Solution of the Burgers’ Equation Using Chelyshkov Polynomials

“…These include the variational iteration method [6], automatic differentiation method [7], finite difference schemes [8], modified cubic B-splines collocation and cubic hyperbolic Bspline method [9,10], least-squares quadratic B-spline finite element method [11], finite element collocation method [12], mixed finite volume element methods [13], sinc-Galerkin method [14], and Laplace transform decomposition method [15]. Orthogonal polynomials play a vital role in solving a wide range of mathematical problems, including differential and integro-differential equations [16][17][18][19][20][21][22][23][24][25]. Chelyshkov polynomials, a class of orthogonal polynomials, possess properties similar to classical orthogonal polynomials and are associated with hypergeometric functions, orthogonal exponential polynomials, and Jacobi polynomials [26,27].…”