An improved cubic Petrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains

A modified cubic Petrov-Galerkin method for solution of multidimensional unsteady advection-diffusion problems is proposed. The method is based on a weighted residual formulation in which the trial space is piecewise linear and the test space is a special piecewise cubic space. The test functions are formed by adding a symmetric cubic perturbation to the piecewise linear basis functions. A frequency-fitting algorithm is used to determine the appropriate magnitude of the cubic perturbation. With this frequency fitted parameter, the method is shown to be superior to standard Galerkin and Petrov-Galerkin methods. Both one- and two-dimensional numerical results are presented. The two-dimensional results are for a rotating flow field and demonstrate that the method performs very well in nonconstant velocity fields. The numerical results demonstrate that the accuracy of the cubic Petrov-Galerkin method is comparable to that of a quadratic basis function Galerkin method, with less than half the computational effort.