Optimal Upstream Collocation Solution of the One-Dimensional Steady-State Convection-Diffusion Equation

We give herein analytical formulas for the solution of the Hermite collocation discretization of the unforced 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 advanced by employing upstream weighting of the convective term in an optimal way, avoiding both smearing and unwanted oscillations, particularly for large Péclet numbers. Computational examples illustrate the efficacy of using optimal upstream weighting