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

Abstract

Abstract. 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 threedimensional 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.Key words. ODEs on a manifold, phase space method, escape equations, high frequency wave propagation, geodesics, creeping rays, seismic waves, travel-time AMS subject classifications. 53C22, 65N06, 65Y20, 78A05, 78A40 IntroductionWe want to compute trajectories on two-dimensional compact manifolds possibly embedded in a higher-dimensional space. The dynamics of the trajectories we consider are given by systems of ODEs in a phase space. In many problems, we need to compute a large number of trajectories. In other words, the dynamical systems of ODEs need to be integrated for many different initial conditions. Examples include geodesics computation in computational geometry [11], robotics [2] and the theory of general relativity. Our motivation for this comes from high frequency wave propagation problems. We consider the problem of scattering of a time-harmonic incident field by a bounded scatterer D. We split the total field into an incident and a scattered field. The scattered field in the region outside D is given by the Helmholtz equation,

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