Boundary conditions for spherical harmonics expansion of Boltzmann equation

1992 ◽  
Vol 28 (11) ◽  
pp. 995-996 ◽  
Author(s):  
D. Schroeder ◽  
D. Ventura ◽  
A. Gnudi ◽  
G. Baccarani
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Spherical Harmonics ◽  
Spherical Harmonics Expansion

AN INFINITE SYSTEM OF DIFFUSION EQUATIONS ARISING IN TRANSPORT THEORY: THE COUPLED SPHERICAL HARMONICS EXPANSION MODEL

2001 ◽  
Vol 11 (05) ◽  
pp. 903-932 ◽  
Author(s):  
PIERRE DEGOND
Keyword(s):  
Boltzmann Equation ◽  
Diffusion Model ◽  
Spherical Harmonics ◽  
Transport Theory ◽  
Weak Sense ◽  
Uniqueness Result ◽  
Diffusion Operator ◽  
The Boltzmann Equation ◽  
Free Case ◽  
Spherical Harmonics Expansion

In this paper, we derive a diffusion model of "SHE" type from a diffusion approximation of the Boltzmann equation. SHE (or Spherical Harmonics Expansion) models consist of a diffusion equation in the position-energy space. In the present model, the diffusion operator couples the various energy levels, while they stay uncoupled in the standard SHE model, at least in the force-free case. This new model is expected to give a better description of particle transport when inelastic collisions are important. The duality structure of the diffusion operator is underlined. A uniqueness result for the diffusion model is shown. A rigorous proof that the solutions of the Boltzmann equation converge (in a weak sense) towards those of the SHE model is developed in the force-free case.

Simulating the diffraction line profile from nanocrystalline powders using a spherical harmonics expansion

2018 ◽  
Vol 74 (6) ◽  
pp. 640-646
Author(s):  
K. R. Beyerlein ◽  
P. Scardi
Keyword(s):  
Line Profile ◽  
Spherical Harmonics ◽  
Diffraction Line ◽  
Nanocrystalline Powders ◽  
Computationally Efficient ◽  
Profile Function ◽  
Diffraction Line Profile ◽  
Cubic Crystallites ◽  
Crystallite Shape ◽  
Spherical Harmonics Expansion

An accurate description of the diffraction line profile from nanocrystalline powders can be obtained by a spherical harmonics expansion of the profile function. The procedure outlined in this work is found to be computationally efficient and applicable to the line profile for any crystallite shape and size. Practical examples of the diffraction pattern peak profiles resulting from cubic crystallites between 1 and 100 nm in size are shown.

scholarly journals Total Electron Content (TEC) estimation over Pakistan from local GPS network using spherical harmonics

2021 ◽  
Vol 64 (1) ◽  
Author(s):  
Maria Mehmood ◽  
Sajid Saleem ◽  
Renato Filjar ◽  
Najam Naqvi ◽  
Arslan Ahmed
Keyword(s):  
Spherical Harmonics ◽  
Orbit Determination ◽  
Region Of Interest ◽  
Local Model ◽  
Small Scale ◽  
Electron Content ◽  
Total Electron ◽  
Gps Network ◽  
Spherical Harmonics Expansion ◽  
The Impact

Many organizations allow GNSS users to access Global Ionosphere Maps (GIMS). However, the TEC estimates derived from GIMs are of insufficient quality to describe small-scale TEC variations over Pakistan. In this paper, the first local TEC map over Pakistan for the year 2019, derived from a regional GPS network, is presented. Spherical harmonics expansion is employed to estimate TEC with the spatial resolution of latitude 0.2° x longitude 0.2° and temporal resolution of 5 minutes. The impact of changing the degree/order of harmonics is assessed and it is determined that harmonic expansion up to 6 degrees is sufficient for estimating accurate TEC map for the region of interest. We have demonstrated that the TEC maps of Pakistan generated by local model conform better to the GIM by Center of Orbit Determination (CODE) (RMS = 5.83) as compared to International Reference Ionosphere (IRI-2016) (RMS = 7.18). We found that the TEC estimated by the local model shows a better correlation to measured TEC; CODE-GIM overestimated TEC, while IRI-2016 underestimates it. Moreover, it was observed that TEC peaks during noon (1100-0100 LT) and Equinox (April). The residuals of local TEC estimates with respect to TEC obtained from CODE- GIM indicate the inaccuracy of CODE-GIM over the region of Pakistan: highest deviation of TEC from local model with respect to CODE –GIM was observed in April (RMS = 8.73) and minimum in October (RMS = 2.78). We have also analyzed the performance of our maps in geomagnetically disturbed days. The research presented in this paper will contribute towards the ionosphere study over Pakistan, where limited research is available currently.

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