NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

2006 ◽  
Vol 17 (11) ◽  
pp. 1563-1577 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Peclet Number ◽  
Eigenvalue Problems ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Péclet Number ◽  
Velocity Models ◽  
Advection Diffusion ◽  
The Stability

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

STABILITY AND ACCURACY OF LATTICE BOLTZMANN SCHEMES FOR ANISOTROPIC ADVECTION-DIFFUSION EQUATIONS

2009 ◽  
Vol 20 (04) ◽  
pp. 633-650 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Benchmark Problems ◽  
Relaxation Parameters ◽  
Advection Diffusion ◽  
The Stability ◽  
Stability And Accuracy

The stability of the numerical schemes for anisotropic advection-diffusion equations derived from the lattice Boltzmann equation with the D2Q4 particle velocity model is analyzed through eigenvalue analysis of the amplification matrices of the scheme. Accuracy of the schemes is investigated by solving benchmark problems, and the LBM scheme is compared with traditional implicit schemes. Numerical experiments demonstrate that the LBM scheme produces stable numerical solutions close to the analytical solutions when the values of the relaxation parameters in x and y directions are greater than 1.9 and the Courant numbers satisfy the stability condition. Furthermore, the numerical solutions produced by the LBM scheme are more accurate than those of the Crank–Nicolson finite difference scheme for the case where the Courant numbers are set to be values close to the upper bound of the stability region of the scheme.

Accurate numerical schemes based on the lattice Boltzmann methods for multi-dimensional diffusion equations

2014 ◽  
Vol 25 (04) ◽  
pp. 1350104
Author(s):  
Shinsuke Suga
Keyword(s):  
Fourth Order ◽  
Lattice Boltzmann ◽  
Three Dimensional ◽  
Relaxation Parameter ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Velocity Models ◽  
Dimensional Diffusion ◽  
Error Terms

We propose accurate explicit numerical schemes based on the lattice Boltzmann (LB) method for multi-dimensional diffusion equations. In LB schemes, the velocity models D2Q9 and D2Q13 are used for two-dimensional equations and D3Q19 and D3Q25 for three-dimensional equations. We introduce free parameters that characterize the weight of the equilibrium distribution functions to reduce numerical errors. Consistency analysis through the fourth-order Chapman–Ensgok expansion of the distribution functions gives an approximate diffusion equation with error terms up to fourth-order. The relaxation parameter and weight parameters are determined so that second-order error terms are eliminated in the approximate equation. Stability analysis shows that we can find a relaxation parameter so that each of the presented schemes is stable for given diffusion coefficients and discretizing parameters. Numerical experiments for the isotropic and anisotropic benchmark problems show that the presented schemes derived from the velocity models D2Q13 and D3Q25 are useful for numerical simulations of practical problems governed by two- and three-dimensional diffusion equations, respectively. In particular, schemes in which the value of the relaxation parameter is set to be 1 demonstrate a fourth-order accuracy under the stability condition.

scholarly journals On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations

2020 ◽  
Vol 378 (2175) ◽  
pp. 20190400
Author(s):  
Stephan Simonis ◽  
Martin Frank ◽  
Mathias J. Krause
Keyword(s):  
Lattice Boltzmann ◽  
Soft Matter ◽  
Collision Operator ◽  
Diffusion Equations ◽  
Lattice Boltzmann Methods ◽  
Discrete Velocity ◽  
Velocity Models ◽  
Advection Diffusion ◽  
Convergence Results ◽  
Relaxation Systems

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

scholarly journals Discrete effect on single-node boundary schemes of lattice Bhatnagar–Gross–Krook model for convection-diffusion equations

2019 ◽  
Vol 31 (01) ◽  
pp. 2050017
Author(s):  
Liang Wang ◽  
Xuhui Meng ◽  
Hao-Chi Wu ◽  
Tian-Hu Wang ◽  
Gui Lu
Keyword(s):  
Boundary Condition ◽  
Relaxation Time ◽  
Lattice Boltzmann ◽  
Diffusion Equations ◽  
Bgk Model ◽  
Distance Ratio ◽  
Single Node ◽  
Convection Diffusion ◽  
Velocity Models ◽  
Discrete Effect

The discrete effect on the boundary condition has been a fundamental topic for the lattice Boltzmann method (LBM) in simulating heat and mass transfer problems. In previous works based on the anti-bounce-back (ABB) boundary condition for convection-diffusion equations (CDEs), it is indicated that the discrete effect cannot be commonly removed in the Bhatnagar–Gross–Krook (BGK) model except for a special value of relaxation time. Targeting this point in this paper, we still proceed within the framework of BGK model for two-dimensional CDEs, and analyze the discrete effect on a non-halfway single-node boundary condition which incorporates the effect of the distance ratio. By analyzing an unidirectional diffusion problem with a parabolic distribution, the theoretical derivations with three different discrete velocity models show that the numerical slip is a combined function of the relaxation time and the distance ratio. Different from previous works, we definitely find that the relaxation time can be freely adjusted by the distance ratio in a proper range to eliminate the numerical slip. Some numerical simulations are carried out to validate the theoretical derivations, and the numerical results for the cases of straight and curved boundaries confirm our theoretical analysis. Finally, it should be noted that the present analysis can be extended from the BGK model to other lattice Boltzmann (LB) collision models for CDEs, which can broaden the parameter range of the relaxation time to approach 0.5.

Characterization of stationary mixing patterns in a three-dimensional open Stokes flow: spectral properties, localization and mixing regimes

2009 ◽  
Vol 639 ◽  
pp. 291-341 ◽  
Author(s):  
M. GIONA ◽  
S. CERBELLI ◽  
F. GAROFALO
Keyword(s):  
Finite Length ◽  
Peclet Number ◽  
Scaling Laws ◽  
Eigenvalue Problems ◽  
Operating Conditions ◽  
Creeping Flow ◽  
Scaling Analysis ◽  
Dominant Eigenvalue ◽  
Péclet Number ◽  
The Real

This article analyses stationary scalar mixing downstream an open flow Couette device operating in the creeping flow regime. The device consists of two coaxial cylinders of finite length Lz, and radii κ R and R (κ < 1), which can rotate independently. At relatively large values of the aspect ratio α = Lz/R ≫ 1, and of the Péclet number Pe, the stationary response of the system can be accurately described by enforcing the simplifying assumption of negligible axial diffusion. With this approximation, homogenization along the device axis can be described by a family of generalized one-dimensional eigenvalue problems with the radial coordinate as independent variable. A variety of mixing regimes can be observed by varying the geometric and operating parameters. These regimes are characterized by different localization properties of the eigenfunctions and by different scaling laws of the real part of the eigenvalues with the Péclet number. The analysis of this model flow reveals the occurrence of sharp transitions between mixing regimes, e.g. controlled by the geometric parameter κ. The eigenvalue scalings can be theoretically predicted by enforcing eigenfunction localization and simple functional equalities relating the behaviour of the eigenvalues to the functional form of the associated eigenfunctions. Several flow protocols corresponding to different geometric and operating conditions are considered. Among these protocols, the case where the inner and the outer cylinders counter-rotate exhibits a peculiar intermediate scaling regime where the real part of the dominant eigenvalue is independent of Pe over more than two decades of Pe. This case is thoroughly analysed by means of scaling analysis. The practical relevance of the results deriving from spectral analysis for fluid mixing problems in finite-length Couette devices is addressed in detail.

Tomographic inversion by matrix transformation

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. VE35-VE38 ◽  
Author(s):  
Jonathan Liu ◽  
Lorie Bear ◽  
Jerry Krebs ◽  
Raffaella Montelli ◽  
Gopal Palacharla
Keyword(s):  
Linear System ◽  
Seismic Velocity ◽  
Synthetic Data ◽  
Velocity Model ◽  
Matrix Transformation ◽  
Uniform Grid ◽  
Complex Structures ◽  
Tomographic Inversion ◽  
Velocity Models ◽  
The Stability

We have developed a new method to build seismic velocity models for complex structures. In our approach, we use a spatially nonuniform parameterization of the velocity model in tomography and a uniform grid representation of the same velocity model in ray tracing to generate the linear system of tomographic equations. Subsequently, a matrix transformation is applied to the system of equations to produce a new linear system of tomographic equations using nonuniform parameterization. In this way, we improved the stability of tomographic inversion without adding computing costs. We tested the effectiveness of our process on a 3D synthetic data example.

scholarly journals The stability of stratified spatially periodic shear flows at low Péclet number

2015 ◽  
Vol 27 (8) ◽  
pp. 084104 ◽  
Author(s):  
Pascale Garaud ◽  
Basile Gallet ◽  
Tobias Bischoff
Keyword(s):  
Peclet Number ◽  
Shear Flows ◽  
Péclet Number ◽  
Spatially Periodic ◽  
The Stability
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