A finite difference scheme for integrating the Takagi-Taupin equations on an arbitrary orthogonal grid

TL;DR

This paper introduces a new finite difference scheme for solving the Takagi-Taupin equations on arbitrary orthogonal grids, improving flexibility and maintaining accuracy for X-ray diffraction simulations.

Contribution

A novel finite difference method that enables integration on orthogonal grids using Fourier interpolation, enhancing computational flexibility for dynamical diffraction calculations.

Findings

Achieves second order convergence similar to traditional methods.

Maintains comparable error levels on similarly dense grids.

Allows integration on arbitrary orthogonal grids, not just sheared ones.

Abstract

Calculating dynamical diffraction patterns for X-ray topography and similar x-ray scattering-imaging techniques require the numerical integration of the Takagi-Taupin equations. This is usually performed with a simple second order finite difference scheme on a sheared computational grid with two of the axes aligned with the wave vectors of the incident and scattered beams respectively. This dictates, especially at low scattering angles, an oblique grid of uneven step-sizes. Here we present a finite difference scheme that carries out this integration in slab-shaped samples on an arbitrary orthogonal grid by implicitly utilizing Fourier interpolation. The scheme achieves the expected second order convergence and a similar error to the traditional approach on similarly dense grids.