Evaluation of quadrature schemes for the discrete ordinates method

After reviewing the presently available quadrature schemes for the discrete ordinates method, the accuracy of different schemes is analyzed and evaluated. It is shown from a comprehensive error analysis that the moment conditions have to satisfied not only for the principal coordinates directions, as it is mostly carried out, but for any arbitrary test direction. Among the schemes with approximately 50 discrete ordinates the DCT-020-2468 quadrature was found to give the best accuracy. The highest accuracy among all schemes is achieved by the LC-11 quadrature which requires 96 discrete ordinates. This scheme is rarely used up to date and deserves more attention for high accuracy predictions.     

Orthonormal bases generated by Cuntz algebras

We show how some orthonormal bases can be generated by representations of the Cuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third Cantor set.     

Investigation of quadrature imbalance compensation algorithm for coherent 6PolSK-QPSK

6PolSK-QPSK is a promising modulation format in optical fiber communication. Because of the damage suffered during the transmission and reception, a series of algorithms are needed to be adopted to recover the original data. We proposed a novel quadrature imbalance compensation algorithm based on the data statistical properties. Simulation results show that the quadrature imbalance can be well compensated with the proposed algorithm.     

三阶非线性发展方程的线性化有理逼近方法

研究了三阶非线性发展方程的初边值问题的解。采用基于Sinc函数的微分求积法发展了线性化有理逼近方法。通常的配点法不适用于上述三阶问题的求解。本文把提出的方法用于求解KdV方程,取得了良好的效果。     

A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

     

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.     

Quadrature demodulation technique for self-mixing interferometry displacement sensor

A new, to the best of our knowledge, signal processing method based on quadrature demodulation technique is presented for laser diode self-mixing interferometry(LDSMI) displacement sensor. Phase modulation of the laser beam is obtained by an electro-optic modulator (EOM) in the external cavity. Detection of the target's displacement can be easily achieved by sampling the interference signal at those times which satisfied certain conditions. The major advantage of the technique is that it does not involve any complicated calculation and insensitive to the sampling error. Experimental results show that the proposed method can effectively improve the displacement measurement resolution of the laser diode self-mixing displacement sensor to a few nanometers.     

有界变差函数的加权Ostrowski型不等式的改进及其应用(英文)

本文改进了有界变差函数的加权Ostrowski型不等式,并给出了其在求积公式中的应用.     

On the best interval quadrature formulae for classes of differentiable periodic functions

In this paper we solve the problem about optimal interval quadrature formula for the class W^rF of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearrangements of averaged monosplines is proved.     

A dynamical strategy for approximation methodsUne stratégie dynamique pour les méthodes d'approximation

The numerical result provided by an approximation method is affected by a global error, which consists of both a truncation error and a round-off error. Let us consider the converging sequence generated by successively dividing by two the step size used. If computations are performed until, in the convergence zone, the difference between two successive approximations is only due to round-off errors, then the global error on the result obtained is minimal. Furthermore its significant bits which are not affected by round-off errors are in common with the exact result, up to one. To cite this article: F. Jézéquel, C. R. Mecanique 334 (2006).