- Advances in the Theory of Nonlinear Analysis and its Application
- Vol: 7 Issue: 1
- Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and...
Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions
Authors : Zineb LAOUAR, Nouria ARAR, Abdelhamid TALAAT
Pages : 133-147
Doi:10.31197/atnaa.1139533
View : 270 | Download : 175
Publication Date : 2023-03-31
Article Type : Research Article
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.Keywords : Spectral method, Galerkin, Robin\'s conditions, Advection-Diffusion equation, Gauss-Quadrature, Difference scheme, Crank Nicolson scheme.