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The goal of this paper is to describe a novel parallel high-resolution 3D numerical method for the solution of high-frequency electromagnetic wave propagation. The sequential numerical method was developed by the first author in 2014. The discussed parallel algorithm will be used later by the authors to computationally simulate data for the solution of the inverse problem of imaging mine-like targets. Thus the solution of the forward problem presented in this paper is a necessary prelude to the future solution of a related inverse problem. In this paper, land mines are modeled as small abnormalities embedded in an otherwise uniform media with an air-ground interface. These abnormalities are characterized by the electrical permittivity and the conductivity, whose values differ from those of the host media. The main challenge in the calculation of the scattered electromagnetic signal in these settings is the requirement of solving the Helmholtz equation for high frequencies. This is excessively time-consuming using standard direct solution techniques. A high-resolution and scalable numerical procedure for the solution of this equation is described in this paper. The kernel of this algorithm is a combination of a second, fourth or sixth order compact finite-difference scheme and a preconditioned Krylov subspace approach. Both fourth and sixth order compact approximations for the Helmholtz equation are considered to reduce approximation and pollution errors, thereby softening the point-per-wavelength constraint. The coefficient matrix of the resulting system is not Hermitian and possesses positive as well as negative eigenvalues. This represents a significant challenge for constructing an efficient iterative solver. In our approach, this system is solved by a combination of Krylov subspace-type method with a direct parallel FFT-type preconditioner. The resulting numerical method allows a natural and efficient implementation on parallel computers. Numerical results for realistic ranges of parameters in soil and mine-like targets confirm the high efficiency of the proposed parallel iterative algorithm.
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