A new approach to numerical integration by means of steady-state flux in a slab

The definite integral is generally interpreted geometrically as an “area”. An alternate interpretation as a steady-state “flux” through a unit slab is derived, which leads to a new method of numerical integration. The usual sum of a large number of approximate areas is replaced by the flux through a “single” increment.The method involves the solution of a system of linear finite difference equations. The coefficient matrix is tri-diagonal and is solved efficiently by the Thomas algorithm. During the iterative process the coefficients are determined by simple quadrature schemes applied to each increment.Error analysis revealed that an expression could be derived for the roundoff error associated with the final Thomas iteration. It is shown that the roundoff error is smallest when the matrix coefficients a_k\S>1. The method is shown to be superior to the classical methods due to its simplicity and tolerance for variable increment size. In addition, a new function is determined which is useful in diffusion studies. Numerical data are presented confirming these results.