Macroscopic model and truncation error of discrete Boltzmann method

Abstract

A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman–Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection–diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection–diffusion–reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier–Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.

Introduction

As a mesoscopic numerical approach, the Boltzmann method has been proven to be a useful mathematical tool to simulate lots of complicated physical phenomena including fluid flow, interface problems, particle and suspension flows for its simplicity and flexibility in numerical aspects [1], [2], [3], [4], [5], [6]. It is constructed based on kinetically microscopic view to mimic macroscopically physical process in the way that the macroscopic fluid behavior is the natural response of assembled microscopic particles [7], [8]. Meticulous particle distributions in velocity phase space are put forward to represent fluid properties from their collective effects. Basically, only streaming and collision operators for the particle distribution evolution are required in practicing a simulation. The macroscopic dynamics will be recovered by assembling microscopic kinetic equation with appropriate equilibrium particle distribution model and collision relaxation time scale. That is, computational parameters regarding particle distribution and collision relaxation time in the discrete Boltzmann method must be properly set up to recuperate the accordingly macroscopic model.

Chapman–Enskog expansion is a viable technique in relating the discrete Boltzmann method to its macroscopic counterpart. It analyzes the particle distribution evolution with two time scales stemmed from flow convection and diffusion processes. The evolution equation is subsequently expanded with a power series of a small parameter (ε) delegating the time scale ratio. Besides, particle distribution is assumed to approximately differ from its equilibrium state by a power series of this small parameter. Macroscopically equivalent equations will be ensued from agglomerating all components of particle distributions and equalizing various order of this pretentious parameter in the resultant evolution equation [4], [9], [10], [11], [12]. For example, in computational fluid dynamics (CFD), the Euler equations are obtained from the lowest order and the Navier–Stokes equations by putting in additional diffusion terms deduced from the second lowest order term. Furthermore, as suggested by Latt [13], with appropriate correction terms in collision process, it will substantially increase the stability and accuracy in numerical simulations which is revealed in the resulting macroscopic model. Although its effectiveness to attain the macroscopic model can be assured, concerns regarding its meaning and interpretation consequently arise [14], [15], [16], [17]:

• Execution of the Chapman–Enskog expansion is essentially based on presumption of two time scales exposed in the continuous model. However, this is not a prerequisite requirement to implement the discrete Boltzmann method. That is, from the numerical viewpoint, the discrete Boltzmann method may be regarded as a mere mathematical technique to handle particle streaming and collision operators.

• The intricate meaning of the introduced small parameter (ε) accounting for the time scale ratio between different physical processes requires rigorous phenomenal explanation. In general, it is interpreted as the Knudsen number to measure the ratio between particle mean-free path and system characteristic size. However, Watari [17] pointed out that it is natural to use time step (Δt) as ε which consequentially leads to an alternative in deriving macroscopic model. Without a stringent evidence on the interpretation of ε, it will become conceptually perplexed in the applications of the Boltzmann method.

• In a convection–diffusion problem, two time scales are used to depict the characteristic times associated with the convection and diffusion processes, respectively. This small parameter (ε) is also taken up to signify the ratio between these time scales. Nevertheless, it becomes ambiguous to explain its physical meaning if there exists an additional physical process with different characteristic time scale.

• Although macroscopic model can be acquired with the Chapman–Enskog expansion, it is difficult to categorize the associated truncation error adherent to the discrete Boltzmann method. Higher-order terms such as Burnett and super-Burnett hydrodynamics may violate the basic physics behind the Boltzmann equation [15], [18].

• Deviation of particle distribution function from its equilibrium state is expressed as a power series of the same small parameter. The first (ε) and second (${\epsilon }^2$) terms are regarded as those brought about by the convection and diffusion processes, respectively. It is not clearly explained the physical meanings of the higher-order terms in this approximate series.

These ambiguities suggest that a more clearly explained derivation procedure is highly demanded in the applications of discrete Boltzmann method. Therefore, efforts have been devoted to avoid the cumbersome consequences adherent to the Chapman–Enskog expansion. For example, to get rid of the presumption on the asymptotic relationship between temporal and spatial discretization parameters, Holdych et al. [19] successfully developed a recursive application of the lattice Boltzmann method and Taylor series expansion to derive the resulting truncation error. On the other hand, Junk et al. [20], [21] introduced a diffusive scaling ($\mathrm{\Delta }t\approx \mathrm{\Delta }{x}^2$) as well as a regular expansion technique rather than the convective scaling ($\mathrm{\Delta }t\approx \mathrm{\Delta }x$) in the Chapman–Enskog analysis to directly obtain the incompressible Navier–Stokes equations. Convergence of the lattice Boltzmann method for one-dimensional convection–diffusion–reaction equations can then be proven [22]. However, an asymptotic relationship between the particle distribution function and its equilibrium state is assumed in the derivation procedure.

The objective of present study is to propose an alternative technique to derive the macroscopic model of the discrete Boltzmann method without the presumptions in the Chapman–Enskog expansion. Thanks to the linearity introduced by the BGK approximation [23], original formulation describing particle distribution function evolution can be reorganized to its equivalent form for equilibrium state. That is, analyses will be conducted with evolution of equilibrium distribution which is directly related to macroscopic properties. Taylor series expansion incorporating with fundamental algebraic manipulations on the discrete Boltzmann method can then be employed to deduce its macroscopic counterpart. Without the intrusion of artificially multiple time scales and confounding small parameter on the derivation procedure, a more comprehension on the conceptual insight of the discrete Boltzmann method can then be obtained.

Besides this introductory section, the present paper is organized as follows. Analysis is conducted for the one-dimensional convection–diffusion problem in Section 2. Both Chapman–Enskog and present proposition are investigated to depict their essential ingredients for comparison. The present analysis is verified by a numerical experiment performed in Section 3. It emphasizes on the truncation error predicted by present formulation which is difficult to attain in the Chapman–Enskog expansion. In the last section, conclusive remarks drawn from the present study are exhibited. Extensions to deal with convection–diffusion–reaction system, multi-relaxation times in collision process as well as multi-dimensional Navier–Stokes equations are exposed in Appendix.