On certain finite difference schemes for hyperbolic systems

Abstract

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 stability condition. The author is indebted to several staff members at the Couraint Institute of Mathematical Sciences for advice and encouragement, especially Professors R. D. Richtmyer and H. B. Keller. The computations described herein were all carried out on the IBM 7090 at New York University.