On the influence of numerical boundary conditions Original Research Article

Our understanding about the behaviour of numerical solutions for evolutionary convection–diffusion equations is mainly based on analysis of infinite domains situations with stability given by von Neumann analysis. Almost all practical problems involve physical domains with boundaries. For evolution problems with Dirichlet boundary conditions, some algorithms can be used without alteration near a boundary. However, the application of higher order methods such as Quickest or second order upwinding introduces difficulty near an inflow boundary, since for interior points adjacent to the boundary there are insufficient upstream points for the high order scheme to be applied without alteration. For that reason such methods require a careful treatment on the inflow boundary, where additional numerical boundary conditions have to be introduced. The choice of numerical boundary conditions turns out to be crucial for stability. A test problem is described, showing the practical advantages of some numerical boundary conditions versus the others by comparison with an exact solution.