Heinz-Otto Kreiss
Numerical Analysis
School of Computer Science and Communication
KTH
Stockholm
Recently Anders Petersson, Omar Ortiz and myself have developed a rather general theory
for second order hyperbolic systems based on Laplace and Fourier transform, with
particular emphasis on boundary processes corresponding to generalized eigenvalues.
Our theory uses mode analysis and builds upon the theory for first order systems
developed a long time ago. This approach has many desirable properties: 1) Once a
second order system has been Laplace and Fourier transformed it can always be written
as a system of 2n first order pseudo-differential equations. Therefore the above
theory also applies here. 2) We can localize the problem, i.e., it is only necessary
to study the Cauchy problem and halfplane problems with constant coefficients,
3) The class of problems we can treat is much larger than previous approaches based
on "integration by parts". 4) The relation between boundary conditions and boundary phenomena becomes transparent.
I will not bore you with complicated proofs. Instead I will discuss a number of examples.
1) Wave equation (Kreiss, Ortiz, Petersson)
2) Einstein's equation of general relativity (harmonic gage) (Kreiss, Winicour)
3) Elastic wave equation (Kreiss, Petersson)
4) Characteristic boundary conditions (Kreiss, Winicour)