A novel finite difference formulation for differential expressions involving both first and second derivatives

It is shown that the upwind difference scheme of formulating differential expressions, in problems involving transport by simultaneous convection and diffusion, is superior to the central differences scheme, when the local Peclet number of the grid is large. Even better schemes are derived and discussed. It is pointed out that the best finite differences analogues are found by approximating differential expressions as a whole, and that simple (e.g. one‐dimensional) exact solutions form a useful, legitimate and independent source of these optimum algebraic formulae.