On certain finite difference schemes for hyperbolic systems

Abstract

1. Introduction. The purpose of this paper is to describe a few finite difference schemes for the numerical solution of hyperbolic systems of partial differential equations. Our main concern will be the stability conditions for these schemes although some of these schemes may be useful, especially for problems involving supersonic fluid flow. We will describe numerical experiments designed to check the derived stability conditions and also the accuracy of these schemes.In Section 2 we will describe the application of the Crank-Nicholson scheme to hyperbolic systems. This method is unconditionally stable, however it requires the inversion of a block tridiagonal matrix. We describe two modifications which require only the inversion of a scalar tridiagonal matrix. We carry out a stability analysis which shows that these schemes are unconditionally stable only if the matrix of the system is positive definite (supersonic flow in the case of hydrodynamics). Results of numerical computations using all these schemes are described in Section 3. All our results are for problems in one space dimension. These methods can be generalized to two dimensions but we have no analysis to indicate that the generalizations will work. In Section 5 we give the results of fluid dynamics computations using three versions of the Lax-Wendroff difference scheme. The objective here is to determine if it is necessary to write the equations in conservation form in order to obtain good results when the flow contains a shock. In Section 6 an application of the Lax-Wendroff scheme to the Navier-Stokes equations is described. An empirical stability condition is obtained which is a combination of the usual hyperbolic and parabolic conditions. In Section 7 an explicit difference scheme related to the CrankNicholson scheme is given. This is an iterative scheme with a rather peculiar sta-

An Iterative Finite-Difference Method for Hyperbolic Systems

“…In view of the above we decided to use a single stability criterion for all the various cases of different degree of implicity (0) When 0 = | this reduces exactly to the case treated by Gary [1], [2]. We have repeated his calculations for the case of weak shock wave (pi/po = 1.4) and obtained the same result as reported by him.…”

“…The emphasis in this research was on the handling of discontinuities, such as shock waves, and overcoming the post-shock oscillations resulting from nonlinear instabilities. The success of the method is indicated by the monotonie profiles which were obtained for almost all the cases tested.The method is based on an idea which was first examined by Gary [1] and considered by him to be unsatisfactory [2]. It will be shown what modifications are needed to insure stability and monotonicity of the flow property profiles.…”

“…The method is based on an idea which was first examined by Gary [1] and considered by him to be unsatisfactory [2]. It will be shown what modifications are needed to insure stability and monotonicity of the flow property profiles.…”

“…The Differential Equations. The system of partial differential equations under consideration is (1) dW/dt + A-dW/dx = 0.…”

“…I11 the open literature one finds two versions [1], [3], [5], Both forms preserve the integrated conservation law, but in general the resulting difference equation does not satisfy the conservation requirements "in the small." In other words, usually Aj+x/2-AW 9e AF.…”