Communications in Mathematical Sciences

Volume 5 (2007)

Number 3

A multiple-patch phase space method for computing trajectories on manifolds with applications to wave propagation problems

Pages: 617 – 648



Mohammad Motamed

Olof Runborg


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.


ODEs on a manifold; phase space method; escape equations; high frequency wave propagation; geodesics; creeping rays; seismic waves; travel-time

2010 Mathematics Subject Classification

53C22, 65N06, 65Y20, 78A05, 78A40

Full Text (PDF format)