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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.

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 |