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This paper reviews the real-space multislice method of numerically computing diffracted electron amplitudes and phases. It is found that the number in of terms of propagator expanded is dependent only on the number N of sampling points and the slice thickness epsilon. The practical procedures for accurate use of this method are discussed in detail.

Petrov-Galerkin method is proposed for solving nonlinear reaction-diffusion equations. The concept of ordered pair of supersolution and subsolution is introduced. A monotone iteration is provided for solving the resulting problem. The convergence of the Petrov-Galerkin scheme is discussed. The numerical results show the advantages of such method.

A dynamical convergence criterion for the real-space multislice method for simulating high-resolution electron microscopy images has been derived. This criterion imposes a practical limitation in determining dynamically the number of terms of the propagation operator P(R) expanded in the subsequent dynamical diffraction calculations concerned with the phase grating selected, the wavefunction and its variation during propagating after choosing the sampling interval Δ and the slice thickness σ. It has been found that, as long as the dynamic criterion is satisfied, much better agreement with the results using a fast Fourier transform multislice method can be obtained.

This paper reports a study of the quasi-real-space (QRS) method. It is an intermediate procedure between the real-space (RS) and multi-slice (MS) methods. With this technique, most of the numerical calculations are done in RS except that normalization is imposed beyond some threshold value, e.g. σ|φ|2 > 1[sdot]01, where φ is the wavefunction of the electron. It is also numerically shown that truncation of the propagation operator exp (λeδ) to second order in the RS method can lead to computational divergences and how they can be avoided by using the QRS method. Finally, results calculated by the QRS method are compared with those obtained by other existing slice methods. The QRS method gives results similar to the conventional MS calculation with competitive computational speed.

A δ-e criterion in the real-space (RS) method for simulating high-resolution electron microscope images has been derived. This condition imposes a practical limitation in choosing the sampling interval δ and the slice thickness e for the RS method. The validity of the new criterion for the RS image simulation in high-resolution electron microscopy has been rigorously examined. It has been found that when the δ-e condition is satisfied, the RS method gives results similar to the conventional multislice method with a saving in computational time but avoiding the computing divergence which can arise in the RS method.