Approximate-Analytical Solution of the Diffusion, Convection and Reaction Problem in Homogeneous Media

Abstract

In convex homogeneous domains, the diffusion, convection and reaction (DCR) problem may be solved by applying Green’s function solution technique. When this technique is applied, the solution to the DCR problem consists of the sum of a set of integrals whose integrands involve the Green’s function. The Green’s function is singular at the upper limit of the time integral and is nonuniformly convergent at the boundaries of the domain. Due to this behaviour, numerical evaluation of the integrals is prohibitively expensive and in some cases, the integrals are incorrectly evaluated. The method presented in this work circumvents all the difficulties inherent with the numerical quadrature of the intergrals and in preliminary case studies (in rectangular coordinates) has reduced the required computation time by up to five orders of magnitude while increasing the accuracy of the results by as much as eight orders of magnitude. The method involves transforming the function in the integrand, which multiplies the Green’s function, into a series of Legendre polynomials. The integral of the product of the Green’s function and Legendre polynomials can be evaluated analytically. This produces both a rapid and accurate evaluation of the integral and subsequently the solution to the DCR problem.