Improved exponential discretization method for convection-diffusion equation

The convective-diffusion equations are very common in applied mechanics.One-dimensional convective-diffusion equation can be given aswhere φ is the unknown function to be solved, such as conceniration, temperature, etc.,t, the time; x, the coordinate; D, the diffusive constant, and V is the velocity of fluidflow.

The convective-diffusion equations([1-3]) are very common in applied mechanics. One-dimensional convective-diffusion equation can be given as partial derivative phi/partial derivative t = (partial derivative/partial derivative x) (D x partial derivative phi/partial derivative x) - V partial derivative phi/partial derivative x, where phi is the unknown function to be solved, such as concentration, temperature, etc.; t, the time; x, the coordinate; D, the diffusive constant, and V is the velocity of fluid flow. The convective-diffusion equation is parabolic, but when V is large it has hyperbolic characteristics. When the equation is discretized by the central difference or Galerkin FEM (finite element method), and if the local Peclet number, Pe = V x Delta x/D, exceeds 2, the solution phi will have some fluctuation behaviour. So the upwind finite difference or finite element schemes are applied. An exponential function interpolation scheme has been developed by Spalding et al. based on which the implemented computer program SIMPLE has been widely spread. However, such a discretization method can be further improved, which is the purpose of the present note, where D and V are considered contants. Only the discretization of coordinate x is given here; the extension to two or three dimensions is easy.