Gauss-Hermite quadrature

The Gauss-Hermite quadrature in the third direction of velocity phase space is the same as for velocity component v, and thus can be formed as a tensor product of the quadrature weights and abscissas in the second direction. However, in practice, it is more computationally efficient to rewrite the terms in Eq. (6.68) using symmetry to eliminate the dependence on the third velocity component. 

Note that, because the Gauss-Hermite quadrature weights and abscissas depend only on the number of points, they need be computed only once if M and M2 are constant (which will usually be the case). 

As in Eq. (8.102), the weights and abscissas in this expression come from (i) the ECQMOM, (ii) Gauss-Laguerre quadrature, and (iii) Gauss-Hermite quadrature. Likewise, Mi and M2 can be chosen as large as needed in order to minimize the quadrature error in the spatial fluxes and drag terms. 

The lattice Boltzmann equation (LBE) is obtained as a dramatic simplification of the Boltzmann equation, (20.1) along with its associated equations, (20.2-20.4). In particular, it was discovered that the roots of the Gauss-Hermite quadrature used to exactly and numerically represent the moment integrals in (20.4), corresponds to a particular set of few discrete particle velocity directions c in the LBE [11]. Eurther-more, the continuous equihbrium distribution (20.3) is expanded in terms of the fluid velocity m as a polynomial, where the continuous particle velocity is replaced by the discrete particle velocity set ga obtained as discussed above, i.e., becomes f 1. After replacing the continuous distribution function / =f x, t) in (20.1) by the discrete distribution function/ =f x, (20.1) is integrated by considering... 

For the two-dimensional case, third-order Gauss-Hermite quadrature leads to the nine-speed LBE model with the discrete velocities... 

The multivariate GHQ rule extends the univariate /n-point set to the n-dimen-sional point set by the tensor product mle [29, 30]. It is exact for all polynomials of the form xj X2 - x with ltotal number of points Np = increases exponentially with the dimension n. Hence, it is hard to use for high dimensional problems. To alleviate this problem, the sparse Gauss-Hermite quadrature can be used [32]. In this paper, the conventional Gauss-Hermite quadrature is used since the dimension of this problem is three. 

Arasaratnam I et al (2007) Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. Proc IEEE 95 953-977... 

Jia B et al (2011) Sparse Gauss-Hermite quadrature filter with application to spacecraft attitude estimation. J Guidance Control Dyn 34 367-379... 

Typically, the quadrature formulae that can be used are either of the non-nested t3q>e such as the Gauss-Legendre and Gauss-Hermite quadrature rules or nested t)q)e. A brief discussion on a couple of nested quadrature rules are presented next. 

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