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Keywords = Weierstrass spectrum

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20 pages, 2021 KiB  
Article
Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited
by Călin-Ioan Gheorghiu
Symmetry 2021, 13(5), 761; https://doi.org/10.3390/sym13050761 - 27 Apr 2021
Cited by 2 | Viewed by 1873
Abstract
In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift [...] Read more.
In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered. Full article
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16 pages, 11263 KiB  
Article
Comparative Study on Numerical Calculation of 2-D Random Sea Surface Based on Fractal Method and Monte Carlo Method
by Gengkun Wu, Chuanxi Liu and Yongquan Liang
Water 2020, 12(7), 1871; https://doi.org/10.3390/w12071871 - 30 Jun 2020
Cited by 2 | Viewed by 2268
Abstract
Based on fifty one groups of data on direction distribution measured from buoy sites, several important spectrum parameters including distribution characteristics of the measured data’s spectrum, the Wen’s direction spectrum and the Donelan function are analyzed from the perspectives of standard deviation of [...] Read more.
Based on fifty one groups of data on direction distribution measured from buoy sites, several important spectrum parameters including distribution characteristics of the measured data’s spectrum, the Wen’s direction spectrum and the Donelan function are analyzed from the perspectives of standard deviation of directional distribution function and statistical results. Then, a numeric calculation model based on the Monte Carlo method is proposed in this work. At the same time—based on Weierstrass self-affine fractal function—numeric simulation of random sea surface is conducted by modifying the bilateral power law. The analysis of the numeric calculation results under different wind directions, speeds and fetches proves that both methods can be adopted for different water directional distributions and target spectrum models. In addition, this study compares the characteristic wave within different distribution frequency domains in the main wave direction and in its orthogonal direction. It is proved that the fractal method cannot fully reflect the anisotropy of gravity wave and tension wave in the superposition direction, however, it can maintain the similarity of overall energy part with the rough length of the waves. Moreover, there are still limitations of the fractal method in the numeric modeling of undeveloped sea surface. Full article
(This article belongs to the Special Issue Observation and Modeling of Coastal Morphodynamics)
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