A High-Order, Fast Algorithm for Scattering Calculation in Two Dimensions

We present a high-order, fast, iterative solver for the direct scattering calculation for the Helmholtz equation in two dimensions. Our algorithm solves the scattering problem formulated as the Lippmann-Schwinger integral equation for compactly supported, smoothly vanishing scatterers. There are two main components to this algorithm. First, the integral equation is discretized with quadratures based on high-order corrected trapezoidal rules for the logarithmic singularity present in the kernel of the integral equation. Second, on the uniform mesh required for the trapezoidal rule we rewrite the discretized integral operator as a composition of two linear operators: a discrete convolution followed by a diagonal multiplication; therefore, the application of these operators to an arbitrary vector, required by an iterative method for the solution of the discretized linear system, will cost N² log(N) for a N-by-N mesh, with the help of FFT. We will demonstrate the performance of the algorithm for scatterers of complex structures and at large wave numbers. For numerical implementations, GMRES iterations will be used, and corrected trapezoidal rules up to order 20 will be tested.