## Computational high frequency wave propagation.(English)Zbl 1049.65098

In this large paper, the authors review own and other achievements, mostly of the last ten years (giving some ninety references from 1949 until today), in the computation of waves of very high frequency. This problem has applications for example in underwater acoustics, seismology, optical and microwaves, and radar cross section problems.
The main philosophy is to compute not (the fast changing) solution values, but (the slowly changing) phase functions and (maximal) amplitudes. The classical technique is ray tracing here (which is a method of characteristics for the eiconal equation having the advantage of the superposition principle), and the many numerical examples presented in the paper compare with this technique. Ray tracing has, however, drawbacks since the rays may cross (where, e.g., the minimum travel time is difficult to establish) or leave empty considerable parts of the solution domain.
In this paper, after showing in the part about the mathematical background the way from the wave equation to geometrical optics and to the geometrical theory of diffraction, mainly such methods are considered which relay on the Liouville equation (for particle density). This approach leads to systems of hyperbolic equations and related numerical techniques (Godunov, Lax-Friedrichs,…).
In detail wave front methods and moment methods are considered. The wave front is defined by the Liouville equation and tracked, e.g., by the segment projection method developed by the authors. Included in the consideration is the treatment of singularities like caustics where the solution becomes multivalued and the amplitude unbounded. The authors also discuss broadly the application of moment methods to the Liouville equation. Here, the resulting system of (nonstrictly) hyperbolic equations is closed, e.g., by assuming that at most (say) two rays cross at any given point in time and space. In this part of the paper, the authors give proofs for a series of theoretical results on the flux and source functions of those hyperbolic systems as well as for their solutions.

### MSC:

 65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35L05 Wave equation 35L45 Initial value problems for first-order hyperbolic systems 78A05 Geometric optics 78A45 Diffraction, scattering 78M05 Method of moments applied to problems in optics and electromagnetic theory
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