Stability Of Lattice Boltzmann Method Research Articles

Linear and brute force stability of orthogonal moment multiple-relaxation-time lattice Boltzmann methods applied to homogeneous isotropic turbulence.

Multiple-relaxation-time (MRT) lattice Boltzmann methods (LBM) based on orthogonal moments exhibit lattice Mach number dependent instabilities in diffusive scaling. The present work renders an explicit formulation of stability sets for orthogonal moment MRT LBM. The stability sets are defined via the spectral radius of linearized amplification matrices of the MRT collision operator with variable relaxation frequencies. Numerical investigations are carried out for the three-dimensional Taylor-Green vortex benchmark at Reynolds number 1600. Extensive brute force computations of specific relaxation frequency ranges for the full test case are opposed to the von Neumann stability set prediction. Based on that, we prove numerically that a scan over the full wave space, including scaled mean flow variations, is required to draw conclusions on the overall stability of LBM in turbulent flow simulations. Furthermore, the von Neumann results show that a grid dependence is hardly possible to include in the notion of linear stability for LBM. Lastly, via brute force stability investigations based on empirical data from a total number of 22 696 simulations, the existence of a deterministic influence of the grid resolution is deduced. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Impact of collision models on the physical properties and the stability of lattice Boltzmann methods.

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method (LBM) with two amending functions for solving partial differential equations (PDEs) arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O(N), where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

Improvement of the instability of compressible lattice Boltzmann model by shock-detecting sensor

Recently, lattice Boltzmann method (LBM) has drawn attention as an alternative and promising numerical technique for simulating fluid flows. The stability of LBM is a challenging problem in the simulation of compressible flows with different types of embedded discontinuities. This study, proposes a complementary scheme for simulating inviscid flows by a compressible lattice Boltzmann model in order to improve the instability using a shock-detecting procedure. The advantages and disadvantages of using a numerical hybrid filter on the primitive or conservative variables, in addition to, macroscopic or mesoscopic variables are investigated. The study demonstrates that the robustness of the utilized LB model is improved for inviscid compressible flows by implementation of the complementary scheme on mesoscopic variables. The validity of the procedure to capture shocks and resolve contact discontinuity and rarefaction waves in well-known benchmark problems is investigated. The numerical results show that the scheme is capable of generating more robust solutions in the simulation of compressible flows and prevents the formation of oscillations. Good agreements are obtained for all test cases.

Enhancement of the stability of lattice Boltzmann methods by dissipation control

Artificial dissipation is a well known tool for the improvement of stability of numerical algorithms. However, the use of this technique affects the accuracy of the computation. We analyse various approaches proposed for enhancement of the Lattice Boltzmann Methods’ (LBM) stability. In addition to some previously known methods, the Multiple Relaxation Time (MRT) models, the entropic lattice Boltzmann method (ELBM), and filtering (including entropic median filtering), we develop and analyse new filtering techniques with independent filtering of different modes. All these methods affect dissipation in the system and may adversely affect the reproduction of the proper physics. To analyse the effect of dissipation on accuracy and to prepare practical recommendations, we test the enhanced LBM methods on the standard benchmark, the 2D lid driven cavity on a coarse grid (101×101 nodes). The accuracy was estimated by the position of the first Hopf bifurcation points in these systems. We find that two techniques, MRT and median filtering, succeed in yielding a reasonable value of the Reynolds number for the first bifurcation point. The newly created limiters, which filter the modes independently, also pick a reasonable value of the Reynolds number for the first bifurcation.

Entropic Lattice Boltzmann Method for high Reynolds number fluid flows

The Lattice Boltzmann Method (LBM) has emerged as a computationally efficient and increasingly popular numerical method for simulating complex fluid flow. However, the standard LBM has always suffered from severe numerical instability. In this paper, we mainly focus on Entropic Lattice Boltzmann Methods (ELBM) for high Reynolds number fluid flow. ELBM, which is derived from H-theorem, enhances the numerical stability of LBM. The compliance of ELBM with H-theorem makes ELBM much more stable than the standard LBM, but the computational cost increases because of solving a non-linear equation. We propose a new optimisation strategy to improve the efficiency of ELBM, and also implement some numerical tests to validate the computational stabilities and correctness.

Stability of lattice Boltzmann methods in hydrodynamic regimes

The Von Neumann linear stability theory as applied by Sterling and Chen [J. Comput. Chem. 123, 196 (1996)] to lattice Boltzmann numerical methods is revisited and extended. A simplifying assumption made by these authors (on the character of the most unstable mode) is abandoned and immediate refinements on their stability results are attained. The inadequacy of uniform background flow, as a general point of expansion, is evident from simulations of simple shear waves. The stability theory is consequently extended to address the destabilizing role of ``background'' shear. To this end, exact time-dependent solutions of the nine-velocity Bhatnagar-Gross-Krook (BGK) lattice Boltzmann model (LB9) are derived and used as expansion points for the stability theory. Calculations reveal both physical and nonphysical instabilities, the former being interpreted via classical inviscid stability theory and the latter forming an empirical instability criterion (fitting better at small values of the viscosity), $N<{R}^{0.56}$, where $N$ is the number of mesh points in the shearing direction and $R$ is the flow Reynolds number. This is interestingly close to the Kolmogorov-Batchelor-Kraichnan inertial range cutoff ${R}^{1/2}$ for two-dimensional isotropic turbulence. In this case, stability seems to require at least the spatial resolution required for accuracy. We also note that the particular class of solutions found above for the LB9 model can be compared directly to corresponding solutions of the Navier-Stokes equations. It is demonstrated that setting $\ensuremath{\tau}=1$, where $\ensuremath{\tau}$ is the relaxation time of the BGK collision operator, provides optimal accuracy in time. This observation may be relevant to current studies as letting $\ensuremath{\tau}\ensuremath{\rightarrow}1/2$ appears to be a common technique aimed at lowering the viscosity and thereby increasing the Reynolds number of LB simulations.