Iterative path expansion for Helmholtz scattering with Neumann boundary conditions

This study presents an iterative numerical scheme for Helmholtz scattering with Neumann boundary conditions, modeling scattering from bounded convex bodies as a sum over propagation paths. The solution is expressed as the sum of the incident wavefield, specular reflection, and edge diffraction contributions. Recasting the Neumann series representation of the nth-order solution into tensor form provides a path interpretation and formally connects iterative path-tracing approaches to the known diffraction operator solution of the scattering problem. The study applies Nyström discretization to the nested diffraction integral, yielding reusable path diffraction coefficients. An iterative scheme is proposed that efficiently explores this path-tensor structure through successive expansions. The absolute value of the wavefield associated with each path serves as the ordering key in a max heap prioritization scheme. Numerical scattering experiments on the unit cube demonstrate rapid convergence. Relative L2-norm differences in the Dirichlet trace drop below 5%, 3.5%, and 3% after 10 000 iteration steps for wavenumbers k=2,4,6 m-1, respectively, when compared to results from direct boundary element formulations. For the k=2 m-1 case, which shows the largest trace error after 10 000 iteration steps, relative L2-norm errors of <2.5% and relative L∞-norm errors of <4% in the domain are observed.