Extensive analysis of the lattice Boltzmann method on shifted stencils

Abstract

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)]. From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity. Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.

Semi-Lagrangian lattice Boltzmann method for compressible flows

“…A stability analysis [64,65] could clarify the exact limits of the scheme. An obvious way to further increase the Mach number is to incorporate statically shifted [22,66] or dynamically shifted lattices [25]. Future works should thoroughly investigate the path to extend the velocity sets to higher velocities in three dimensions, while keeping the number of discrete velocities lower than 125.…”

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

“…Hence, the above equilibrium states do recover some of the third- and fourth-order moments of the Maxwell–Boltzmann distribution. While fourth-order equilibrium moments do not have any impact on the isothermal macroscopic behaviour (but only on the numerical stability [11,12,76,78]), third-order ones are required to recover the correct viscous stress tensor [see equation (2.2)]. Indeed, considering the standard second-order equilibrium [by imposing Mpqeq=0 for p + q ≥ 3 in equation (2.9)], macroscopic errors related to the viscous stress tensor ΔΠ pq are normalΔΠ20=∂xfalse(M30eqfalse)+∂yfalse(M21eqfalse), normalΔΠ11=∂xfalse(M21eqfalse)+∂yfalse(M12eqfalse)2emand2emnormalΔΠ02=∂xfalse(M12eqfalse)+∂yfalse(M03eq<...>…”

“…Later, several authors performed LSA of the LBM, which significantly improved the understanding of its numerical properties and stability limits [8,9,16,60,67,68,75,102]. In addition, a number of works investigated the behaviour of collision models operating in a particular moment space (raw [8,61,70], Hermite [9,16,64], central [70,72] and central Hermite [77]) or based on a regularization step [11,76–78]. These studies were usually presented in their own framework, and using different methodologies, which prevents any general comparison of the results obtained for each collision model.…”

“…Interestingly, this upper stability limit can be shown to depend on both the lattice and the type of equilibrium (polynomial, entropic, etc.). In addition, it does not seem to be impacted by either the collision model [11,12,76], or the numerical discretization [74].

Figure 4Mach number (Ma) corresponding to the linear stability limit of the D2Q9-BGK-DVBE, for various angles ϕ of the mean flow and ν = 10 −6 .

…”

“…In fact, by adopting a different type of equilibrium, e.g. the entropic formulation obtained by minimizing the H-functional under the constraint of density and momentum conservation [134], this limitation could be alleviated but at the cost of part of the physics [76]. To improve both the physical and numerical properties of LBMs, one could include correction terms for the velocity-dependent errors that remain in the definition of the viscous stress tensor [26,47,77,85,92–95].…”