Finite differences for the convection-diffusion equation : on stability and boundary conditions
The solution of convection-diffusion problems is a challenging task for numerical methods because of the nature of the governing equation, which includes a non-dissipative component and a dissipative component. Once the convection-diffusion equation is discretised, it is usual to observe oscillations in the computed solution regardless of whether these might be expected in the original physical situation. Mostly these oscillations are the result of numerical instability. This thesis centres on this fundamental difficulty: the numerical stability of finite difference discretisation of a convection-diffusion equation. The existence of an exact evolution operator for the constant coefficient convection diffusion problem is the framework we use to derive new finite difference schemes in one and two dimensions and also, when a high-order scheme is considered, to derive numerical boundary conditions. The influence of numerical boundary conditions on the stability of a general scheme is one of the main themes. The stability analysis is done mostly by using the von Neumann method and the matrix method. The Godunov-Ryabenkii theory is also applied to the one dimensional case. In two dimensions we deduce different forms of second-order (Lax-Wendroff) schemes and third-order (Quickest) schemes. We apply some of those schemes to a Navier-Stokes problem by running experiments to illustrate the practical stability region, showing how results from a simpler case presented in previous chapters carry over to the more complex case.