Multidimensional optimization of finite difference schemes for Computational Aeroacoustics

Three novel finite-difference time-domain (FDTD) methods based on the split-step (SS) scheme with high-order accuracy are presented, which are proven to be unconditionally stable. In the first novel method, symmetric operator and uniform splitting are adopted simultaneously to split the matrix derived from the classical Maxwell's equations into six sub-matrices. Accordingly, the time step is divided into six sub-steps. The second and third proposed methods are obtained by adjusting the sequence of the sub-matrices deduced in the first method, so all the novel methods presented in the paper have similar formulations, of which the numerical dispersion errors and the anisotropic errors are lower than the alternating direction implicit finite-difference time-domain (ADI-FDTD) method, the initial SS-FDTD method and the modified SS-FDTD method based on the Strang-splitting scheme. Specifically, for the second method, corresponding to a certain cell per wavelength (CPW), there is a Courant number value making the numerical anisotropic error to be zero, while in the third novel method, corresponding to a certain Courant number value, there exists a CPW making the numerical anisotropic error to be zero. In order to demonstrate the high-order accuracy and efficiency of the proposed methods, numerical results are presented.

Three New Unconditionally-Stable FDTD Methods With High-Order Accuracy

Based on the concept of isotropic centered finite differences, this work generalizes the spatial order of accuracy of the 2D and 3D isotropic discrete gradient operators to a higher order. A suitable methodology is used to obtain a set of equations from which it is possible to deduce stencil weights to achieve numerical approximations of both high order spatial and high order isotropic gradients. We consider that the suggested discretization will be useful for enhancing the quality of the results in various scientific fields. The spatial order ( $$S$$ S ) controls the spatial order of accuracy of the gradient norm and direction, while the isotropic order ( $$I$$ I ) controls, in some situations, the spatial order of accuracy of the gradient direction. A useful list of the stencil weights needed to construct different high order spatial and isotropic gradients is given. Numerical tests show that the numerical spatial orders of accuracy of the gradient approximation are the same as those predicted theoretically. Also, to illustrate the benefit of the new discretizations, some simulations with a multiphase lattice Boltzmann model are presented. Then, a series of benchmarks comparing various efficient convolution algorithms used to compute function or image gradients is presented. Different platforms implemented on CPUs and GPUs are studied, namely: plain MATLAB; the Jacket plugin for MATLAB; and CUDA. The results show situations in which substantial computational speedup can be obtained with CUDA and the Jacket plugin for MATLAB versus MATLAB on a CPU. Examples of 2D and 3D gradient computations using convolution products performed with our code are available for download as electronic supplementary material.

High Order Spatial Generalization of 2D and 3D Isotropic Discrete Gradient Operators with Fast Evaluation on GPUs

A new split-step finite difference time domain (SS-FDTD) method with high-order accuracy is presented, which is proven to be unconditionally stable and has four substeps. The numerical dispersion error and the numerical anisotropic error of the proposed method are reduced than the alternating direction implicit finite difference time domain method, the conventional SS-FDTD method and the SS-FDTD method based on the Strang-splitting scheme. The proposed method has new splitting forms, which is different from the SS-FDTD method based on the exponential evolution operator. At each time step, an important aspect is that the proposed method produces 33% reduction of the total number of arithmetic operators than the SS-FDTD method based on the exponential evolution operator. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 562–565, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24100

High‐order accurate FDTD method based on split‐step scheme for solving Maxwell's equations

The split-step finite difference time domain (SS-FDTD) method characterised by unconditional stability is becoming an important numerical method in computational electromagnetics. Proposed is a new high-order accurate unconditionally stable SS-FDTD method, which is derived from the exponential evolution operator. Compared with the conventional SS-FDTD method, the numerical dispersion of the new method is greatly reduced.

High-order accurate split-step FDTD method for solution of maxwell's equations

Comparison of Taylor finite difference and window finite difference and their application in FDTD

Phase noise is an important index in evaluating the performance of millimeter wave (MMW) frequency source. Because of the high frequency, it is difficult to measure its phase noise directly. So it is very necessary to find new methods for estimating it effectively and easily. In this paper, the main factors affecting phase noise of MMW PLL frequency source are analyzed, and then a new method to estimate the phase noise is presented, which is based on the comparison of the phase noise of microwave phase-locked frequency source with phase-locked intermediate frequency in MMW phase-locked loop. In order to demonstrate the validity of this method of phase noise estimate, it is applied to estimate the phase noise of 95GHz double PLL frequency synthesizer. The result shows that the theoretical estimate value is well coincident with the experimental value.

Phase Noise Analysis and Estimate of Millimeter Wave PLL Frequency Synthesizer

Usually, 1D finite difference based on the Taylor series expansion theorem is used to approximate the spatial ideal partial-differential operator (IPDO) using conventional FDTD methods. Such a treatment is simple, but its severe numerical dispersion is untenable. In this paper, a kind of 3D isotropic finite difference (IFD) is introduced and a new FDTD method, called IFD-FDTD, is presented. A numerical-dispersion analysis shows that it is superior to Yee's conventional FDTD method. In addition, through the analysis of stability, it is found that its stability condition is the same as that of Yee's FDTD method. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 46: 381–384, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20993

Xianjing Zhang

Papers

Comparison of Taylor finite difference and window finite difference and their application in FDTD

Phase Noise Analysis and Estimate of Millimeter Wave PLL Frequency Synthesizer

3D low-dispersion IFD-FDTD based on 3D isotropic finite difference