ITERATIVE TECHNIQUES FOR ROUGH-SURFACE SCATTERING PROBLEMS

The problem of wave scattering at a randomly rough surface gives rise to a non-Hermitian linear system, which corresponds to the discretization of a linear Fredholm integral equation of the first or second kind. In this paper, we describe the use of various iterative methods for the solution of the real linear non-symmetric system, corresponding to the complex system of the scattering problem, and the regions of convergence of the methods in terms of the basic parameters of the problem. It has been found that Generalized Conjugate Gradient method is the only approach valid for very rough surfaces, h/l > 1.0 say (here hll is the ratio of the RMS surface height to surface correlation length). The regions of validity for the iterative methods in terms of convergence and energy conservation are also found and it is shown that this depends not only on hll and kl (here k is the wavenumber of the incident field) but also on the density of mesh points in the discretization. Some convergence properties are discussed by analyzing the spectral radius of the iterative matrix. It is found for small h/l that the spectral radius is linearly proportional to hll and approximately proportional to root kl for large kl. It is also found numerically that when h/l . root kl is greater than one, the iterative matrix for the linear integral equation of the second kind treated here has spectral radius greater than one, and therefore, the Generalized Conjugate Gradient method is not guaranteed to converge since the corresponding linear real system of the second kind may not have positive definite symmetric part. Finally, we present a comparison of iterative techniques for the linear integral equations of the first and second kinds, and find that when such iterative methods are to be employed, it is more efficient to use the second kind integral equation formulation than the first kind formulation.