Extensions of a second order high resolution explicit method for the numerical computation of weak solutions of one dimensonal hyperbolic conservation laws are discussed. The main objectives were (1) to examine the shock resoluton of Harten's method for a two dimensional shock reflection problem, (2) to study the use of a high resolution scheme as a post-processor to an approximate steady state solution, and (3) to construct an implicit in the delta-form using Harten's scheme for the explicit operator and a simplified iteration matrix for the implicit operator.
The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit schemes is emphasized. Some examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit schemes, and unconditionally stable schemes. HEN finite-difference schemes are used to solve initial boundary-value problems for the equations of fluid dynamics, it is well known that most methods require more conditions than those required by the governing partial differential equations. These additional conditions for the finite-difference equations are often called numerical conditions. The conditions cannot be imposed arbitrarily but are determined, in general, using interior information, for example, by extrapolation or uncentered approximations. In this paper, any procedure used to provide a condition will be called a boundary scheme/' Whatever schemes are used for the conditions, it is a common practice to assume that the scheme has a local effect and will not affect the solution globally. During the early 1970s, Kreiss,1'2 Osher,3 Gustafsson et al.,4 Varah,5 and Gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with schemes included. Since then, further progress has been made in the theory of linear difference approximations for initial boundary-value problems of the hyperbolic and parabolic type.7'12 Because improper treatment of the conditions can lead to instability and inaccuracy, even though we start with a stable interior scheme (i.e., scheme for the interior points), it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and schemes. Surveys of recent developments and extensive bibliographies are included in papers by Coughran13 and Yee.14 The purpose of this paper is to examine the applicability to practical calculations (for nonlinear gasdynamic problems) of recent theoretical stability analyses of implicit difference approximations for initial boundary-value problems of the hyperbolic type. As computations have progressed, the use of the conservative form of the gasdynamic equations has gained popularity. For physical reasons it is sometimes desirable to specify conditions in the nonconservative variables and to compute with conservative variables in the interior. We will consider the additional complications introduced by this procedure.
This program determines the supersonic flowfield surrounding three-dimensional wing-body configurations of a delta wing. It was designed to provide the numerical computation of three dimensional inviscid, flowfields of either perfect or real gases about supersonic or hypersonic airplanes. The governing equations in conservation law form are solved by a finite difference method using a second order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically. The flowfield between the body and outermost shock is treated in a shock capturing fashion and therefore allows for the correct formation of secondary internal shocks . The program operates in batch mode, is in CDC update format, has been implemented on the CDC 7600, and requires more than 140K (octal) word locations.
