R. F. Warming

An implicit finite-difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation- law form. The algorithm is second-order- time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross derivatives are evaluated explicitly. However, the algorithm is unconditional ly stable and, although a three-time-lev el scheme, requires only two time levels of data storage. The algorithm is constructed in a form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensiona l shock boundary-layer interaction problem.

Explicit second-order upwind difference schemes in combination with spatially symmetric schemes can produce larger stability bounds and better numerical resolution than symmetric schemes alone. However, if conservation form is essential, a special operator is required for transition between schemes. An operational approach has been devised for deriving transition operators so that strict conservation and local consistency are maintained. Various aspects of hybrid schemes are studied numerically for model linear and nonlinear equations. To demonstrate the utility of combining two different algorithms, MacCormack's explicit, noncentered, second-order method is combined with a completely upwind version, and numerical solutions of the Euler equations are obtained for two-dimensional, transonic flows with embedded supersonic regions and shock I. Introduction 1T4 this paper we consider the application of explicit Isecond-order, one-sided or upwind, difference schemes for the numerical solution of hyperbolic systems in conservation-law form = 0