| 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. This paper presents a basic review of the spectral collocation method. The derivation is driven with an example of the approximation to the solution of a 1D Helmholtz equation. A Matlab code modeling two fluid dynamics problems is then given. First, the classic two-dimensional Graetz problem is simulated and compared to an analytical solution, a finite difference formulation and a published series solution. An implementation that includes the effects of axial conduction is then compared to the classic series solution for low Peclet number flow, as well as several published results. Finally, two-dimensional laminar diffusion in a tube is modeled and compared to a published analytical solution. The application of spectral collocation to these problems is unique to this study and the results suggest that significant speedup can be achieved in other areas. In addition, the results are in excellent agreement with published data and the Matlab code provides an example of a simple yet effective pseudospectral method implementation. |
