High-order boundary integral equation solution of high frequency wave   scattering from obstacles in an unbounded linearly stratified medium
  We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. $(\Delta + E + x_2)u(x_1,x_2) = 0$ where $E$ is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 100 nodes, with an effort that is independent of the frequency parameter $E$. By combining with high-order Nystrom quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop. 
Alex. H. Barnett, Bradley J. Nelson, J. Matthew Mahoney
math.NA