Utilization of the method of characteristics to solve accurately two-dimensional transport problems by finite elements

A new finite element method is presented for the solution of two-dimensional transport problems. The method is based on a weighted residual formulation in which the method of characteristics is combined with the finite element method. This is achieved by orienting sides of the space-time elements joining the nodes at subsequent time levels along the characteristics of the pure advection equation associated with the transport problem. The method is capable of solving numerically the advection--diffusion equation without generating oscillations or numerical diffusion for the whole spectrum of dispersion from diffusion only through mixed dispersion to pure convection. The utility and accuracy of the method are demonstrated by a number of examples in two space dimensions and a comparison of the numerical results with the exact solution is presented in one case. A very favourable feature of the method is the capability of solving accurately advection dominated transport problems with very large time steps for which the Courant number is well over one.