High order finite difference numerical methods for time-dependent convection-dominated problems

We continue to investigate a family of fully discrete finite difference implicit methods already proposed for the numerical solution of one-dimensional hyperbolic systems of conservation laws. In this paper the extension of the Euler schemes to the resolution of convection-dominated convection-diffusion equations is considered. All the properties of numerical schemes are called upon in order to specify conditions on the parameters of the family. The truncation error analysis leads to conditions on the order of accuracy and the development of the equivalent differential equation provides guidelines for optimization of the dispersion and the diffusion errors. The classical Von Neumann method is applied to assess the stability of the schemes which is guaranteed with no restriction on the time step. A wide series of computational experiments is carried out to illustrate and validate the behavior and the capability of the schemes. The numerical results demonstrate that the proposed family has good performance in stability and accuracy.