Boundary conditions for suppressing rapidly moving components in hyperbolic systems, 1

This work is concerned with hyperbolic systems of partial differential equations for which certain of the associated propagation speeds are a great deal larger than the other propagation speeds. Our goal is to find boundary conditions which prevent rapidly moving waves from entering the given spatial domain. Conditions of this type are desirable in certain numerical computations arising in meteorology. In order to find these conditions, we first transform the given system to an approximate diagonal form in such a way that each of the new dependent variables can be identified as a slow, incoming fast, or outgoing fast component of the solution. We then find local boundary conditions which suppress the incoming fast part. We consider only linear systems. These methods are applied in detail to the linearized shallow water equations. The results of numerical tests of various boundary conditions are included. AMS(MOS) subject classifications. 35L50, 65M99 1. Introduction. Hyperbolic partial differential equations are characterized by the fact that they propagate information at finite speed. For first order hyperbolic systems there may be several such propagation speeds, each corresponding to an eigenvalue of the principal symbol of the system. In the present work we consider systems for which the various speeds can have substantially different magnitudes. Systems of this type are sometimes said to have "multiple time scales." Examples of such systems arise in the study of fluid dynamics. The shallow water equations and the Euler equations of gas dynamics admit certain modes associated with the movement of the fluid and certain other modes associated with the propagation of gravity waves and sound waves, respectively. If these waves move at speeds which are considerably greater than rate of flow of the fluid, then these systems have two time scales. In this paper we consider initial-boundary value problems for such systems. The goal is to find boundary conditions which prevent rapidly moving high-frequency waves from entering the given spatial domain. That is, we will try to identify the portion of the solution which is entering the region at the fast speed, and we will then attempt to set this part of the solution equal to zero at the boundary. The purpose of these conditions is to prevent excessive errors in the boundary data from propagating rapidly into the interior during numerical computations. In 2 of this paper we describe a physical problem in which boundary conditions of this type would be useful, and at the end of that section we mention a connection between this work and the construction