Chebyshev, Legendre, Hermite and Other Orthonormal Polynomials in D Dimensions

Doria, Mauro M.; Coelho, Rodrigo C. V.

doi:10.1016/s0034-4877(18)30040-5

Cited by 3 publications

(4 citation statements)

References 19 publications

(33 reference statements)

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“…Here, the normalization factor is the same as for the Hermite polynomials in D-dimensions [24,54], where we define δ i 1 ···i N | j 1 ··· j N ≡ δ i 1 j 1 · · · δ i N j N + all permutations of j's and δ i j is the Kronecker's delta. The weight functions used to build the models in this paper will be discussed in the next section (Eqs.…”

Section: Relativistic Polynomialsmentioning

confidence: 99%

Fully dissipative relativistic lattice Boltzmann method in two dimensions

et al. 2018

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In this paper, we develop and characterize the fully dissipative Lattice Boltzmann method for ultra-relativistic fluids in two dimensions using three equilibrium distribution functions: Maxwell-Jüttner, Fermi-Dirac and Bose-Einstein. Our results stem from the expansion of these distribution functions up to fifth order in relativistic polynomials. We also obtain new Gaussian quadratures for square lattices that preserve the spatial resolution. Our models are validated with the Riemann problem and the limitations of lower order expansions to calculate higher order moments are shown. The kinematic viscosity and the thermal conductivity are numerically obtained using the Taylor-Green vortex and the Fourier flow respectively and these transport coefficients are compared with the theoretical prediction from Grad's theory. In order to compare different expansion orders, we analyze the temperature and heat flux fields on the time evolution of a hot spot.

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Section: Relativistic Polynomialsmentioning

confidence: 99%

Fully dissipative relativistic lattice Boltzmann method in two dimensions

et al. 2018

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“…Properties of the above tensors, such as their number of terms, is discussed in more details Ref. [31]. Using the Chapman-Enskog expansion [32,1], the macroscopic equations and the transport coefficients for a fluid governed by a given EDF f eq can be calculated from the discrete Boltzmann equation,…”

Section: Expansion Of the Equilibrium Distribution Functionmentioning

confidence: 99%

“…The explicitly derivation of the coefficients by imposing the orthonormality of the first five polynomials is carried in Ref. [31]. The coefficients can be summarized as follows:…”

Section: The Generalized Polynomialsmentioning

confidence: 99%

Lattice Boltzmann method for semiclassical fluids

Coelho

Doria

2018

Computers & Fluids

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We determine properties of the lattice Boltzmann method for semiclassical fluids, which is based on the Boltzmann equation and the equilibrium distribution function is given either by the Bose-Einstein or the Fermi-Dirac ones. New D-dimensional polynomials, that generalize the Hermite ones, are introduced and we find that the weight that renders the polynomials orthonormal has to be approximately equal, or equal, to the equilibrium distribution function itself for an efficient numerical implementation of the lattice Boltzmann method. In light of the new polynomials we discuss the convergence of the series expansion of the equilibrium distribution function and the obtainment of the hydrodynamic equations. A discrete quadrature is proposed and some discrete lattices in one, two and three dimensions associated to weight functions other than the Hermite weight are obtained. We derive the forcing term for the LBM, given by the Lorentz force, which dependents on the microscopic velocity, since the bosonic and fermionic particles can be charged. Motivated by the recent experimental observations of the hydrodynamic regime of electrons in graphene, we build an isothermal lattice Boltzmann method for electrons in metals in two and three dimensions. This model is validated by means of the Riemann problem and of the Poiseuille flow. As expected for electron in metals, the Ohm's law is recovered for a system analogous to a porous medium.

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“…But we try here to present a new method based on shifted Legendre polynomials in matrix form using some innovative ideas. There are valuable methods published for the numerical solutions of Fredholm integral equations of the second kind [15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Shifted Legendre Polynomials For Solving Second Kind Fredholm Integral Equations

El-Gohary

Morgan

2021

Menoufia Journal of Electronic Engineering Research

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in this paper, present a computational method for solving Fredholm integral equations of the second kind. The method based on the application of the shifted Legendre polynomials in matrix forms. We create a technique for extracting the Legendre coefficients of each polynomial away so that each Legendre polynomial is rewritten in the form of its coefficient's matrix multiplied by the monomial basis function matrix. This technique significantly reduces the round off errors. By using this technique, the unknown and the data functions are expressed in two forms; each consists of three matrices. The kernel is approximated twice relevant to its two variables so that it is transformed into a form consists of a product of five matrices. By substituting by the approximate unknown function into the left and the right sides of the integral equation, we obtain an algebraic linear system of the equations without applying the collocation points. Moreover, we adapted the Gauss-Quadrature rule in an adjustment form and applied it for computing the resulted integrals. The convergence in the mean of the approximate solution and the kernel are proved. Additionally, the maximum norm error is studied, and it is found equal to zero. Numerical results are obtained for five examples to clarify the simplicity, efficiency, and reliability of the method. The obtained solutions are equal or rapidly converge to the exact solutions.

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Chebyshev, Legendre, Hermite and Other Orthonormal Polynomials in D Dimensions

Cited by 3 publications

References 19 publications

Fully dissipative relativistic lattice Boltzmann method in two dimensions

Fully dissipative relativistic lattice Boltzmann method in two dimensions

Lattice Boltzmann method for semiclassical fluids

Shifted Legendre Polynomials For Solving Second Kind Fredholm Integral Equations

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