Lattice Boltzmann Schemes with Relative Velocities

AbstractIn this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d’Humières. They extend also the Geier’s cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.

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On Triangular Lattice Boltzmann Schemes for Scalar Problems

Communications in Computational Physics ◽

10.4208/cicp.381011.270112s ◽

2013 ◽

Vol 13 (3) ◽

pp. 649-670 ◽

Cited By ~ 1

Author(s):

François Dubois ◽

Pierre Lallemand

Keyword(s):

Lattice Boltzmann ◽

Taylor Expansion ◽

Particle Method ◽

Expansion Method ◽

Internal Vertex ◽

Order Accuracy ◽

Discrete Particle ◽

Discrete Particle Method ◽

Lattice Boltzmann Scheme ◽

Super Convergence

AbstractWe propose to extend the d’Humieres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

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Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

Asymptotic Analysis ◽

10.3233/asy-211692 ◽

2021 ◽

pp. 1-41

Author(s):

François Dubois

Keyword(s):

Partial Differential Equations ◽

Fourth Order ◽

Lattice Boltzmann ◽

Taylor Expansion ◽

Relaxation Times ◽

Nonlinear Case ◽

Order Accuracy ◽

Algebraic Expressions ◽

Formal Expansion ◽

Lattice Boltzmann Scheme

We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this framework. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.

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Solution with Third-order Accuracy for the Electron Distribution Function in Weakly Ionized Gases (or in Intrinsic Semiconductors): Application to the Drift Velocity

Australian Journal of Physics ◽

10.1071/ph810361 ◽

1981 ◽

Vol 34 (4) ◽

pp. 361 ◽

Cited By ~ 10

Author(s):

G Cavalleri

Keyword(s):

Distribution Function ◽

Legendre Polynomials ◽

Taylor Expansion ◽

Electron Distribution ◽

Electron Distribution Function ◽

Mean Free Path ◽

Third Order ◽

Order Accuracy ◽

Weakly Ionized ◽

The Mean

The first four components 10, I" 12 and 13 of the expansion in Legendre polynomials of the electron distribution function I are shown to be of order t:D, et, e2 and e3 respectively, with e = (m/M)'/2 where m and M are the masses of the electron and molecule respectively. This allows the solution of the so-called P3 approximation to the Boltzmann equation applied to a weakly ionized gas (or to an intrinsic semiconductor) in steady-state and uniform conditions and for dominant elastic collisions. However, nonphysical divergences appear in 10 and in the drift velocity W. This can be understood by the equivalence of the Boltzmann-Legendre formulation and the mean free path formulation in which a Taylor expansion is performed around the 'origin', i.e. for a -+ 0, where a = eE/m is the acceleration due to an external electric field E. Indeed, one sees that the expansion under the integral sign (integrals appear in the evaluation of transport quantities) leads to divergent integrals if the expansion is around a = O. Fortunately, it is easy to perform a Taylor expansion around a oft 0 in the mean free path formulation and then to find the corresponding expansion in Legendre polynomials outside the origin. In this way, explicit convergent expressions are found for 10, I" 12, 13 and W, with third-order accuracy in e = (m/M)'/2. This is better than the best preceding expression, that by Davydov-Chapman-Cowling, which has first-order accuracy only (it is the solution of the P, approximation to the Boltzmann equation).

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