Iterative path expansion for Helmholtz scattering with Neumann boundary conditions

Area of Science:

• Computational physics
• Electromagnetics
• Numerical analysis

Background:

• Helmholtz scattering problems with Neumann boundary conditions are crucial in wave physics.
• Modeling scattering from convex bodies requires efficient numerical techniques.
• Existing methods may face challenges with complex geometries and high wavenumbers.

Purpose of the Study:

• To develop an iterative numerical scheme for Helmholtz scattering with Neumann boundary conditions.
• To model scattering from bounded convex bodies using a sum over propagation paths.
• To connect iterative path-tracing to established diffraction operator solutions.

Main Methods:

• An iterative numerical scheme based on a Neumann series representation recast into tensor form.
• Nyström discretization applied to nested diffraction integrals to obtain path diffraction coefficients.
• A max heap prioritization scheme using wavefield absolute values for ordering paths.

Main Results:

• The iterative scheme demonstrates rapid convergence for scattering from a unit cube.
• Relative L2-norm differences in the Dirichlet trace achieved <3% for wavenumbers up to k=6 m-1.
• For k=2 m-1, relative L2-norm errors in the domain were <2.5% and L∞-norm errors <4%.

Conclusions:

• The proposed iterative scheme efficiently explores the path-tensor structure of the scattering problem.
• The method offers a viable and accurate alternative to direct boundary element formulations.
• The approach shows promise for complex wave scattering simulations.