Nonlocal Boundary Conditions for Split-Step Padé Approximations of the Helmholtz Equation With Modified Refractive Index

This research is concerned with the numerical methods for the electromagnetic wave propagation over the Earth's surface. Discrete nonlocal boundary conditions for the Padé parabolic equation are derived. A linear dependence of the modified refractive index on the height outside the computational domain can be taken into account. Thus, the earth-flattening transformation can be used. The proposed approach gives an opportunity to truncate the integration domain without introducing an artificial absorbing layer in the vicinity of the upper boundary. Padé approximations of the exponential propagation operator allow one to carry out the calculations with a rather large step along the longitudinal coordinate. The elaborated method can utilize the two-way parabolic equation approach for scattering problems. Comparison with the results obtained by the split-step Fourier method is given.