A critical review of numerical solution of Navier-Stokes equations

The paper examines the chief problems encountered in obtaining solutions to the partial differential equations of fluid dynamics through numerical integration. The problem of the well-posedness of the differential and the difference equations and the problem of computational stability of a set of difference equations are studied. The application of the von Neumann analysis for the stability of numerical integration of the Navier-Stokes equations is discussed. The application of implicit difference algorithms is examined, with attention given to iterative methods in particular. The problem of conservative difference formulation is discussed, and general methods for heuristic error estimate are outlined. The problems in dealing with shock waves, artificial viscosity and shock-induced oscillations are also reviewed.