A stable and fast method for solving multibody scattering problems via the method of fundamental solutions
This provides a scalable and stable numerical solution for acoustic scattering in physics and engineering, though it is incremental as it builds on existing MFS and scattering matrix techniques.
The paper tackles acoustic multibody scattering problems by computing local scattering matrices for each body and forming a global linear system, resulting in a well-conditioned matrix that scales to many scatterers and is solvable with iterative methods. The method uses the method of fundamental solutions (MFS) for simplicity, maintaining stability despite potential ill-conditioning.
The paper describes a numerical method for solving acoustic multibody scattering problems in two and three dimensions. The idea is to compute a highly accurate approximation to the scattering operator for each body through a local computation, and then use these scattering matrices to form a global linear system. The resulting coefficient matrix is relatively well-conditioned, even for problems involving a very large number of scatterers. The linear system is amenable to iterative solvers, and can readily be accelerated via fast algorithms for the matrix-vector multiplication such as the fast multipole method. The key point of the work is that the local scattering matrices can be constructed using potentially ill-conditioned techniques such as the method of fundamental solutions (MFS), while still maintaining scalability and numerical stability of the global solver. The resulting algorithm is simple, as the MFS is far simpler to implement than alternative techniques based on discretizing boundary integral equations using Nyström or Galerkin.