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A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems. (English) Zbl 1133.65089

We present a multiple-patch phase space method for computing trajectories on two-dimensional manifolds possibly embedded in a higher-dimensional space. The dynamics of trajectories are given by systems of ordinary differential equations (ODEs). We split the manifold into multiple patches where each patch has a well-defined regular parameterization.
The ODEs are formulated as escape equations, which are hyperbolic partial differential equations (PDEs) in a three-dimensional phase space. The escape equations are solved in each patch, individually. The solutions of individual patches are then connected using suitable inter-patch boundary conditions. Properties for particular families of trajectories are obtained through a fast post-processing.
We apply the method to two different problems: the creeping ray contribution to mono-static radar cross section computations and the multivalued travel-time of seismic waves in multi-layered media. We present numerical examples to illustrate the accuracy and efficiency of the method.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
53C22 Geodesics in global differential geometry
78A40 Waves and radiation in optics and electromagnetic theory
86A15 Seismology (including tsunami modeling), earthquakes
78A05 Geometric optics
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
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