Effects of Jacobi polynomials on the numerical solution of the pellet equation using the orthogonal collocation, Galerkin, tau and least squares methods

Abstract

A number of different numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau and least squares (LSQ) methods, are used within the spectral framework to solve a linear reaction–diffusion pellet problem with slab and spherical geometries. The node points are in this work taken as the roots of orthogonal polynomials in the Jacobi family. Two Jacobi polynomial parameters, α and β, can be used to tune the distribution of the roots within the domain. The objective of this paper is thus to investigate the influence of the node point distribution within the domain adopting the weighted residual methods mentioned above. Moreover, the results obtained with the different weighted residual methods are compared to examine whether the numerical approaches show the same sensitivity to the node point distribution. The notifying findings are as follows:(i)Considering the condition number of the coefficient matrices, the different weighted residual methods do not show the same sensitivity to the roots of the polynomials in the Jacobi family. On the other hand, the simulated error obtained adopting the Galerkin, tau and orthogonal collocation methods for different α, β-combinations differ insignificantly. The condition number of the LSQ coefficient matrix is relatively large compared to the other numerical methods, hence preventing the simulation error to approach the machine accuracy.(ii)The Legendre polynomial, i.e., α=β=0, is a very robust Jacobi polynomial giving on average the lowest condition number of the coefficient matrices and the polynomial also give among the best behaviors of the error as a function of polynomial order. This polynomial gives good results for small and large gradients within both slab and spherical pellet geometries.(iii)Adopting the Legendre polynomial, the Galerkin and tau methods obtain favorable lower condition numbers than the orthogonal collocation and LSQ methods.