An Analysis of Some Classes of Petrov-Galerkin and Optimal Test Function Methods

This chapter discusses an analysis of some classes of Petrov-Galerkin and Optimal Test Function methods. Reliable numerical solutions to advection–dominated flow problems are of great importance to many engineering disciplines. Fluid flow at relatively high Reynolds number and convective transport in low-diffusivity fields, are two of the important examples. The development of alternative weighted residual techniques, which give rise to upwind operators in a systematic framework, is one of the most important numerical contributions in this area. Petrov-Galerkin methods and the newly developed Optimal Test Function methods have proven to be very effective for the simulation of advection dominated flows. The chapter develops and analyzes some of these schemes, and proves that, for model one-dimensional steady-state and transient advection diffusion problems, these diverse formulations produce similar or in some cases identical results. The methods considered are: Allen and Southwell difference scheme, quadratic Petrov-Galerkin, streamline upwind Petrov-Galerkin, exponential Petrov-Galerkin and optimal test function methods.