Discretization of a convection-diffusion equation

The unsteady convection-diffusion equation with constant coefficientsadmits an exact solution in the form of a convolution integral,which provides an explicit representation of the evolution operatorthrough one time step. This is used to unify many numericalschemes for the equation, showing interrelationships betweenfinite difference and finite element schemes and presentinga general framework for detailed error analysis. In particular,the upwind scheme, Lax Wendroff, QUICKEST, the ECG schemes andCrank Nicolson are all members of a family that includes powerfulnew schemes. Fourier analysis is used to obtain practical stabilityregions and some insights into accuracy; and the Peano kerneltheorem is also used to derive rigorous error bounds that canbe generalized to irregular meshes.