A fully spectral solution method for parabolic equations

A comparison is made between the performance of the (aliased) Chebȳshev collocation method (CM) and the more recent Galerkin collocation method (GCM), which is a least‐squares collocation method, in solving the laminar, incompressible, steady boundary‐layer equations, which are parabolic in nature. An iterative procedure based on the preconditioned residual minimization method has been used. It is shown that the GCM is superior to the CM on several counts. Unlike the CM, the GCM minimizes the residual uniformly over the entire domain. The global accuracy of the solution is found to be higher in the GCM, at lower grid resolutions. The method also achieves much higher convergence rates. Unlike in the collocation method, the final residual values obtained in the GCM are good indicators of the level of accuracy achieved in the solution. It is highly likely that these results will be repeatable in other systems of parabolic partial differential equations.