FINITE-ELEMENT ANALYSIS OF TIME-DEPENDENT CONVECTION DIFFUSION EQUATIONS (PETROV-GALERKIN).

Abstract

Petrov-Galerkin finite element methods based on time-space elements are developed for the time-dependent multi-dimensional linear convection-diffusion equation. The methods introduce two parameters in conjunction with perturbed weighting functions. These parameters are determined locally using truncation error analysis techniques. In the one-dimensional case, the new algorithms are thoroughly analyzed for convergence and stability properties. Numerical schemes that are second order in time, third order in space and stable when the Courant number is less than or equal to one are produced. Extensions of the algorithm to nonlinear Navier-Stokes equations are investigated. In this case, it is found more efficient to use a Petrov-Galerkin method based on a one parameter perturbation and a semi-discrete Petrov-Galerkin formulation with a generalized Newmark algorithm in time. The algorithm is applied to the two-dimensional simulation of natural convection in a horizontal circular cylinder when the Boussinesq approximation is valid. New results are obtained for this problem which show the development of three flow regimes as the Rayleigh number increases. Detailed calculations for the fluid flow and heat transfer in the cylinder for the different regimes as the Rayleigh number increases are presented.