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.
Publié le : 2007-09-15
Classification:
ODEs on a manifold,
phase space method,
escape equations,
high frequency wave propagation,
geodesics,
creeping rays,
seismic waves,
travel-time,
53C22,
65N06,
65Y20,
78A05,
78A40
@article{1188405671,
author = {Motamed, Mohammad and Runborg, Olof},
title = {A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems},
journal = {Commun. Math. Sci.},
volume = {5},
number = {1},
year = {2007},
pages = { 617-648},
language = {en},
url = {http://dml.mathdoc.fr/item/1188405671}
}
Motamed, Mohammad; Runborg, Olof. A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems. Commun. Math. Sci., Tome 5 (2007) no. 1, pp. 617-648. http://gdmltest.u-ga.fr/item/1188405671/