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 ensure conservation. Results from a one-dimensional shock-tube problem show that expansions are accurately computed and shocks are sharply defined. The algorithm remains stable and accurate for shocks of seemingly unlimited strength and yields an improvement in convergence rate. 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 nonuniform grid.