TIME ACCURATE SOLUTION OF ADVECTION - DIFFUSION PROBLEMS BY FINITE ELEMENTS
The Taylor-Galerkin method, recently proposed for the spatial and temporal discretization of hyperbolic equations, is employed to derive accurate and efficient numerical schemes for the solution of time dependent advection - diffusion problems. Two distinct numerical strategies are discussed: the first is suitable for evolutionary problems, while the second is appropriate for situations in which a steady state is eventually reached. For purely evolutionary problems the Taylor-Galerkin method is applied to the complete advection - diffusion equation and, through modifications of the standard mass matrix, is shown to generate an incremental form of the Crank-Nicolson time stepping method. Such an incremental form also lends itself to the derivation of computationally simple, but accurate, explicit algorithms. To deal with transient situations which evolve towards a highly convective steady state, the global Taylor-Galerkin method is found to be ineffective since it reduces to the standard Galerkin formulation as the temporal term vanishes. For such cases we suggest the use of a splitting-up method in which advection and diffusion are treated separately by appropriate Taylor-Galerkin methods. The proposed methods are illustrated first for linear, constant coefficient equations. They are successively extended to deal with nonlinear and multidimensional problems. Numerical applications indicate their effectiveness.
Bibliographic Reference: COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, VOL. 45 (1984), NOS. 1-3, PP. 123-145
Record Number: 1989123018400 / Last updated on: 1987-01-01
Available languages: en