A nonstandard finite difference scheme for convection–diffusion equations having constant coefficients

In this note we derive, using the subequation method, a new nonstandard finite difference scheme (NSFD) for a class of convection–diffusion equations having constant coefficients. Despite the fact that this scheme has nonlinear denominator functions of the step sizes (even for linear PDEs), it has a couple of favourable properties: it is explicit and due to its construction it reproduces important properties of the solution of the parabolic PDE. This proposed method conserves, by construction, the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the Black–Scholes equation for the valuation of standard options, since it avoids negative values for the calculated prices. Finally, we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature.