Modeling particle diffusion in laminar tube flow with spectral collocation

The spectral collocation method is a numerical approximation technique that seeks the solution of a differential equation using a finite series of infinitely differentiable basis functions. This inherently global technique enjoys an exponential rate of convergence and has proven to be extremely effective in computational fluid dynamics. Despite the initial complexity of understanding spectral collocation, the use of the method is relatively straight forward. Here we present a complete example of applying this method to modeling particle diffusion in laminar tube flow. The included code, written for Octave, highlights the reduction of a partial differential equation into a matrix interpolation using spectral collocation. The results are compared to an analytical solution for accuracy and a finite difference method for performance.