Implementation of Galerkin and moments methods by Gaussian quadrature in advection-diffusion problems with chemical reactions

This work presents a method to solve boundary value problems based on polynomial approximations and the application of the methods of moments and the Galerkin method. The weighted average residuals are evaluated by improved Gauss-Radau and Gauss-Lobatto quadratures, capable to exactly compute integrals of polynomials of degree 2n and 2n + 2 (where n is the number of internal quadrature points), respectively. The proposed methodology was successfully applied to solve stationary and transient problems of mass and heat diffusion in a catalyst particle and of a tubular pseudo-homogeneous chemical reactor with axial advective and diffusive transports. Through the improvement of the usual procedures of numerical quadratures, it was possible to establish a direct connection between the residuals on internal discrete points and the residuals on the boundaries, allowing the method to exactly reproduce the moments and Galerkin methods when applied to linear problems. © 2013 Published by Elsevier Ltd. . Introduction The application of the method of weighted residuals (MWR) consists basically in the approximation of the dependent variables of he problem by expansions series of known functions (called trial functions) with coefficients to be determined. The replacement of this pproximation in the differential equation gives rise to the residual function. Nullifying the weighted average residual functions in the roblem domain, with appropriate weights, it is possible to determine the coefficients of the proposed trial functions. The distinction etween the different methods originates from the choice of the weights to be used in the computation of the weighted average residual unctions. The most widely used methods are: orthogonal collocation method (OCM), method of moments, Galerkin method, and least quare method (Finlayson, 1972; Villadsen & Michelsen, 1978). The selection of trial functions is of great importance to the success of the MWR, because this choice is directly related to the accuracy nd the convergence of numerical solution (Finlayson, 1971, 1972; Finlayson & Scriven, 1966; Snyder, Spriggs, & Stewart, 1964). The use of rthogonal polynomials as trial functions has some advantages, such as the minimization of the residuals maximum magnitude (Villadsen Stewart, 1967). Most applications of this technique use Lagrange polynomial approximation, taking as collocation points the roots of rthogonal polynomials: Finlayson (1971) used the roots of Legendre polynomial, McGowin and Perlmutter (1971) used the roots of the hebyshev polynomial, and Secchi, Wada, and Tessaro (1999), Lefrève, Dochain, Azevedo, and Magnus (2000), Sousa and Mendes (2004), arroso, Henrique, Sartori, and Freire (2006), Damak (2006), Solsvilk and Jakobsen (2012) used the roots of Jacobi polynomials. The use of rthogonal polynomials also extends to majority of the computer packages that make use of MWR, such as PDECOL (Sincovec & Madsen, 979), that uses the Legendre polynomials, COLSYS (Ascher, Christiansen, & Rusell, 1981), that uses the Gauss-Legendre polynomials, WRTools (Adomaitis, 2002; Chang, Adomatis, & Lin, 1999), that uses the Jacobi polynomials, and PDECHEB (Berzins & Dew, 1991), that ses the Chebyschev polynomials. Only few papers present criteria to justify the choice of the Jacobi polynomial parameters ̨ and ˇ. Lefrève et al. (2000) select the values f these parameters to minimize the error of numerical solution. Secchi et al. (1999) select the values of ̨ and ˇ that best approximate he weight of a characteristic function obtained by a self-adjoint form of the system. Solsvilk and Jakobsen (2012) applied orthogonal ollocation, Galerkin, tau and least squares methods, using the Jacobi polynomial with different values of the parameters ̨ and ˇ, in order ∗ Corresponding author at: Universidade de Brasília – UnB, Instituto de Química – IQ, Campus Universitário Darcy Ribeiro, Asa Norte – Brasília – DF, Cep: 70910-000, Caixa ostal 04478, Brazil. Tel.: +55 06131073828; fax: +55 06132734149. E-mail address: [email protected] (E.M. Lemos). 098-1354/$ – see front matter © 2013 Published by Elsevier Ltd. ttp://dx.doi.org/10.1016/j.compchemeng.2013.11.001 t t c u F ( a O a t h m t e m m m J w s a t o h c