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

• Systematic method to solve boundary value problems.
• Appropriate criteria to selection of the orthogonal polynomial.
• Improvement of the usual procedures of numerical Gauss quadratures.
• Direct connection between the residuals on internal points and on the boundaries.
• Mobility of the point where the residue function nullify.

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.