Initial boundary value problems for second order systems of partial differential equations

Heinz-Otto Kreiss
NADA
KTH
Stockholm


Abstract:

Problems concerned with wave propagation in two or three space dimensions are often formulated in terms of systems of wave equations which we have to solve numerically. Examples are Maxwell's equations, elastic wave equations and Einstein's equation of general relativity. We want to solve the initial boundary value problems for t > 0 in a finite domain in space with a smooth boundary. At t = 0 we give initial conditions and boundary conditions which are either Diriclet conditions or relations between normal and tangential derivatives. The most desirable properties for these problems is that there is an energy estimate and that the problem is stable against lower order perturbations. The usual way to prove the existence of an energy estimate is by integration by parts. This is always possible for the Cauchy problem and problems with Diriclet boundary conditions. In these cases the numerical solution poses relatively few difficulties. Physical phenomena like glancing or surface waves lead to derivative boundary conditions which are not maximally dissipative. The energy estimate does not give us a detailed understanding about the behavior of the solution near the boundary which is desireable to develop numerical techniques. By solving a Cauchy problem we can reduce the data such that only the boundary conditions are inhomogeneous. Also, the required stability estimates are such that we can reduce the discussion to halfspace problems. For halfspace problems we will discuss a theory which is based on Fourier and Laplace transform which gives us the desired information about the solution and apply it to the earlier mentioned equations.