Enhancement of the stability of lattice Boltzmann methods by dissipation control

Abstract

Abstract. Artificial dissipation is a well known tool for the improvement of stability of numerical algorithms. At the same time, this affects the accuracy of the computation. We analyze various approaches proposed for enhancement of the Lattice Boltzmann Methods (LBM) stability. In addition to the previously known methods, the Multiple Relaxation Time (MRT) models, the entropic lattice Boltzmann method (ELBM), and filtering (including entropic median filtering), we develop and analyze new filtering techniques with independent filtering of different modes.All these methods affect the dissipation in the system and may adversely affect the reproduction of the proper physics. To analyze 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.1. Introduction. Lattice Boltzmann schemes are a type of discrete algorithm which can be used to simulate fluid dynamics and more [3,12,32]. One of the nicest properties of an LB scheme is that the transport component of the algorithm, advection, is exact. All of the dissipation in the discrete system then occurs due to the relaxation operation. This dissipation occurs at different orders of the small parameter, the time step [28]. The first order gives an approximation to the Navier Stokes equations (with some error terms due to the discrete velocity system). The higher orders include higher space derivatives, similarly to Burnett and super-Burnett type systems. (They differ from the Burnett and super-Burnett terms and may have better stability properties, see a simple case study for the Ehrenfests' collisions in [24]). The form of the dissipative terms in the macroscopic dynamics is driven by the form of the discrete equilibrium, the analogy to the Maxwell distribution, the coefficients of the dissipative terms are selected by the collision operator. In the standard and most basic relaxation operation, the single time BGK relaxation [4], these coefficients cannot be modified independently. One mechanism which is used to stabilize LB schemes is to generalize this to an MRT operator where these coefficients can be varied independently [16,17,13,14,15,25].An MRT relaxation operation involves selecting a new basis to perform relaxation in. This basis must necessarily include the macroscopic moments (which do not vary during relaxation) and also includes elements used to vary the hydrodynamic viscosity. Depending on the size of the velocity set a number of degrees of freedom may remain available. These degrees of freedom can be used to modify the coefficien...

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

“…Nevertheless, it is worth noting that some theoretical work remains to be done in order to properly define the validity range of the approximated minimization problem [19]. Eventually, one can further find in the literature stabilization techniques based on other kinds of dissipation control [20][21][22][23].…”

Numerical investigation of the accuracy, stability, and efficiency of lattice Boltzmann methods in simulating non-Newtonian flow

“…Overrelaxation in subsequent time steps (along the streaming trajectory of f i ) could result in nonphysical oscillations. Considering the effect oscillations of particle distributions will have on macroscopic variables and, consequently, local quasiequilibriums, positive feedback loops can occur causing the system to diverge or "pollute" the system enough to make the results highly nonphysical [21].…”

“…Note that limiting nonequilibrium entropy in LBM is analogous to what flux limiters do in finite difference, finite volume, and finite element methods [9]. A criteria for introducing artificial dissipation that has been used successfully [7][8][9]21] is to define a threshold, θ, such that dissipation is added when:…”

Entropy-Assisted Computing of Low-Dissipative Systems

“…A: The closest analog of the conventional stabilization techniques in the LBM setting is perhaps the method of entropic limiters [27][28][29]. The idea behind is to measure the closeness of the pre-collision state to the corresponding local equilibrium (in the sense of the entropy difference), and to apply equilibration instead of over-relaxation if the difference exceeds a user-defined threshold.…”

“…Various versions of limiters were considered [27][28][29]. The authors of [29] claimed that entropic limiters "perform better" than ELBM.…”