Boundary Approximations for Implicit Schemes for One-Dimensional Inviscid Equations of Gasdynamics

Abstract

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.