Reducing the numerical dispersion of the one-way Helmholtz equation via the differential evolution method

Abstract

This study is devoted to increasing the performance of the numerical methods for solving the one-way Helmholtz equation in large-scale domains. The higher-order rational approximation of the propagation operator was taken as a basis. Computation of an appropriate approximation coefficients and grid sizes is formulated as the problem of minimizing the discrete dispersion relation error. Keeping in mind the complexity of the developed optimization problem, the differential evolution method was used to tackle it. The proposed approach does not require manual selection of the artificial parameters of the numerical scheme. The stability of the scheme is provided by an additional constraint on the optimization problem. Various configurations of the differential evolution method are studied as applied to the considered problem. The advantages of the proposed method has been demonstrated on several classical diffraction problems. Comparisons with the Padé approximation method, rational interpolation and split-step Fourier method are carried out. Theoretical considerations and numerical experiments prove that the proposed method requires much less computational resources than the existing ones. Python 3 implementation of the proposed method is freely available. This work is an extended version of the ICCS-2022 conference paper (Lytaev, 2022).