Optimal hermite collocation solution of a forced convection-diffusion equation

We give herein analytical formulas for the solution of the Hermite collocation discretization of a forced steady-state convection-diffusion equation in one spatial dimension and with constant coefficients, defined on a uniform mesh with Dirichlet boundary conditions. The accuracy of the method is enhanced by optimally employing "upstream weighting" of the convective term and optimally sampling the forcing function, avoiding both the "smearing" effect of numerical diffusion and unwanted oscillations, particularly for large Péclet numbers. Computational examples illustrate the efficacy of our approach.