A shock capturing algorithm for the Navier-Stokes equations

An implicit numerical algorithm for the unsteady Euler and Navier-Stokes equations is presented. This algorithm is based on flux vector splitting to retain the proper direction of information flow in the algorithm's numerical domain of dependence and on a finite volume formulation to insure conservation. Results from a one-dimensional shock tube problem show that expansions are accurately computed and that shocks are sharply defined. The algorithm remains stable and accurate for shocks of seemingly unlimited strength and improvement in convergence rate is obtained. The use of these methods for higher-dimensional viscous flows is discussed and results from a two-dimensional flat plate boundary-layer problem show good accuracy on a two-dimensional non-uniform grid.