Exploit high-frequency scattering asymptotics in a Boundary Element Method. Tested with Matlab 2016b (www.matlab.com), Chebfun 5.6.0 (www.chebfun.org), Sage 6.9 (www.sagemath.org) and Julia 0.4.7-pre (www.julialang.org).
This implementation accompanies the article "On the eigenmodes of periodic orbits for multiple scattering problems in 2D" (in preparation) and the UKBIM conference proceedings pg 99-108 "Coupling modes in high-frequency multiple scattering problems: the case of two circles", both by D. Huybrechs and P. Opsomer in 2017. For the case of two circular scatterers, we compute a Taylor approximation of the equilibrium phase of the density in ray tracing in the Sagemath worksheet LimitCycleTwoCircles.sws. The Taylor coefficients can be computed independently of the wavenumber and incident wave.
This phase is also the eigenvector (V1k9.mat) of a matrix representing a full cycle of reflections, computed in the script makeVb1.m in the folder `matlab'. After adding the path to chebfun, the script phase2Circles.m validates our symbolic results and the geometric interpretation of the stationary point and phase: the latter is the distance to the periodic orbit via an infinite number of reflections at equal angles, where one substracts the distance between the circles at each reflection. The influence of some parameters on the eigenvalue and some eigenvectors of a periodic orbit between two disks is briefly explored in otherEV.m. The script checkPhasePerOrbit.m loads a file created by makeVb1.m for some general geometry and then computes and validates a Taylor approximation of the phase of the mode of the corresponding periodic orbit computed in seriesPhasePerOrbit.m.
There is also part of some code to apply asymptotic compression in 2D in the matlab folder (the full version is www.github.com/popsomer/bempp), and in 3D in the julia folder: see "High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing", submitted, D. Huybrechs and P. Opsomer, 2017.