IMPLICIT CONSERVATIVE CRARACTERISTIC PIODELING SCBEMES POR TEE EULBR EQUATIONS - A NEW APPROACFI Stephen F. Wornom * R83-3qqrb NASA Langley Research Center Hampton, Virginia Abstract An implicit characteristic-modeling solution scheme for the Euler equations is presented. The scheme does not require the governing equations to be written in characteristic variables or the flux terms to be split into positive and negative contributions. For the two-dimensional problem of a shock wave reflecting from a flat plate, this feature and the simple solution algorithm combine to reduce the computational work per mesh point by 40 percent from that required by a standard, central-difference, implicit, solution algorithm. Application of the method to the quasi-one-dimensional nozzle flow equations for subsonic and supersonic flows without shocks shows the method to be well-conditioned for large time steps. I. Introduction The motivation for the flux-vector splitting scheme of Steger and warming1 was to obtain a method more efficient than the implicit central- difference approach used by steger2 for computing flows past airfoils. Since the flux-vector- splitting scheme as applied by Steger would eliminate the costly back-substitution step necessary for the implicit, central-difference, block-tridiagonal solution algorithm, it was felt that the overall work would be reduced. However, as pointed out by Buning and ~ t e ~ e r , ~ increased efficiency was not achieved. The additional work introduced when the flux terms are split into negative and positive contributions is greater than the back-substitution step found in the central-difference, block-tridiagonal solution algorithm. This increase in computational work also applies when conservative flux-vector- or flux- difference-splitting methods are applied in an explicit manner. ~eese~ presented explicit flux- vector-splitting calculations for airfoils which were a factor of three times slower than the explicit central-difference method of ~ameson' to achieve the same accuracy. The additional work was attributed to the increased number of flux calculations and a slower convergence rate. The purpose of this paper is to present a new approach to implicit, conservative, characteristic-modeling schemes. In this paper methods that attempt to model the proper domain of *~esearchScientist, Theoretical Aerodynamics Branch, TAD. Member AIAA. This paper is declared a work of the U.S. Government and therefore is in the public domain. dependence for the Euler equations according to the sign of the characteristic speeds are called - characteristic-modeling schemes. For the one- and two-dimensional cases studied, the present method shows a reduction of about 40 percent in computa- tional work per mesh point and also a faster rate of convergence than a corresponding implicit, three-point, central-difference method. The idea is developed for one-dimensional flows and extended to two-dimensional calculations using an AD1 factored scheme. With this approach, the two-dimensional problem is reduced to two one- dimensional problems and the same ideas can then be applied to each factored sweep. Although the present work is implicit, it may be possible for the current ideas to be implemented explicitly. The key to the present method is the use of two-point central differences for the Euler flux terms as opposed to the widely-used, three-point central differences. This seemingly slight change leads to very important advantages for hyperbolic systems. When a three-point central-difference scheme is applied to a system of first-order differential equations such as the one-dimensional nozzle flow equations, the resulting finite-difference equations admit nonphysical, oscillatory solu- tions. In order to eliminate these nonphysical solutions, fourth-order dissipative terms are added to the system of difference equations. Another undesirable characteristic of the three- point, central-difference scheme is that it requires numerical houndary conditions. Numerical boundary conditions are defined here as those boundary conditions required by the numerical scheme in addition to those required by characteristic theory. As noted previously, Steger and Warming primarily viewed their characteristic-modeling scheme as the more efficient method to obtain solutions to the Euler equations. Their scheme and other similar methods6-'' attempt to account for the proper domain of dependence by associating a flux with each characteristic direction and differencing the flux terns upwind or downwind at a point. These schemes are naturally dissipative and, therefore, eliminate nonphysical solutions. One disadvantage of these methods is that to obtain second-order spatial accuracy, five mesh points are required. These wide bandwidth operators lead to problems near the boundaries. One approach for hyperbolic systems that has been overlooked is the use of an implicit,