DISCRETIZATION OF A CONVECTION-DIFFUSION EQUATION

The unsteady convection-diffusion equation with constant coefficients admits an exact solution in the form of a convolution integral, which provides an explicit representation of the evolution operator through one time step. This is used to unify many numerical schemes for the equation, showing interrelationships between finite difference and finite element schemes and presenting a general framework for detailed error analysis. In particular, the upwind scheme, Lax Wendroff, QUICKEST, the ECG schemes and Crank Nicolson are all members of a family that includes powerful new schemes. Fourier analysis is used to obtain practical stability regions and some insights into accuracy; and the Peano kernel theorem is also used to derive rigorous error bounds that can be generalized to irregular meshes.