Numerical Analysis Group Research Report NA-91/4

K W Morton, I J Sobey

Revised September 1991.

The unsteady convection diffusion equation with constant velocity admits an exact solution in the form of a convolution integral of the concentration at an earlier time and a travelling exponential function. This representation can be put in the form of an evolution operator relating the solution at one time level to that at a previous time level. By using this evolution operator we are able to unify many numerical schemes for the convection diffusion equation, showing inter-relationships between finite difference and finite element schemes and presenting a general framework for further error analysis. In particular the Lax Wendroff approximation arises from the evolution of a quadratic approximation to the initial distribution, and Leonard's QUICKEST scheme comes both from evolving a cubic approximation to the initial distribution and also from a finite element solution using a mixed norm and piecewise linear basis functions. The Peano kernel theorem is used to derive error bounds for these two methods; Fourier analysis is used to obtain practical stability regions and further insight into their accuracy.

Finally, it has long been conjectured that there should be a connection between the optimal test functions that arise in the application of Petrov-Galerkin methods to the steady convection diffusion equation and the test functions arising in finite element approximations to the unsteady advection equation. We prove a direct relationship in the case of a particular mixed norm.

Key words and phrases:

Convection-diffusion, Lax Wendroff, QUICKEST, finite differences, finite elements, optimal test functions

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