A flux reconstruction kinetic scheme for the Boltzmann equation
Introduction
A highly visible direction in the study of computational fluid dynamics (CFD) is the development of high-order numerical schemes. Thanks to the benefits from being intuitive, robust and flexible for implementation, low-order methods, i.e., those which provide a maximum of second order accuracy, are arguably dominant in industrial applications. High-order methods, on the other hand, offer more accurate approximate solutions in a physical system. They are often more efficient than low-order methods in terms of the accuracy achieved per computational degree of freedom, which benefits high-fidelity simulation of intricate flows under a comparable computational cost [1]. However, it is more complex to implement high-order methods and they are basically less robust due to the reduced numerical dissipation. It also remains a focus of research to generate high-order meshes for three-dimensional flow simulations. As a result, the use of high-order methods in academia and industry has so far been limited.
High-order methods have been developed in the context of the finite difference (FD), finite volume (FV) and finite element (FE) formulations. By extending the difference stencils, higher-order finite difference methods can be constructed and it is feasible to construct compact stencils [2]. Such straightforward extensions are restricted to problem domains with regular geometry only [3]. The finite volume methods can handle complex geometries in design, and a series of high-order extensions have been developed with regular and irregular geometries [4], [5], [6], [7]. However, the reconstructions in FV methods are mostly based on cell-averaged values, resulting in non-compact stencils.
The thriving finite element methods provide an alternative to design high-order methods. The discontinuous Galerkin (DG) method is obviously one of the most studied high-order FE algorithms [8], [9], [10], [11], which originates from the work on neutron transport problem by Reed and Hill [12]. The basic idea of the DG methods lies in the unified consideration of spatial discretization and spectral decomposition. Within each element, the solutions are represented via polynomial basis functions and are allowed to be discontinuous across cell boundaries, which encourages the method to capture sharp wave structures that arise in fluid mechanics. Thanks to the in-cell polynomials, it is straightforward to extend the DG methods to arbitrary order of accuracy for smooth solutions. As a special case of DG methods, the nodal DG scheme employs Lagrange polynomials as basis functions to interpolate solutions between distinct nodal points [13]. Such idea is implemented similarly in another class of algorithms named the spectral difference (SD) methods [14], [15], but based on the differential form of governing equations.
Huynh's work on the flux reconstruction (FR) approach provides profound insight into constructing high-order methods for any advection-diffusion type equation [16]. It establishes a general framework, where many existing approaches such as the nodal DG and spectral difference methods can be understood as its particular cases. Jameson used the FR formulation to prove that the SD method is uniformly stable in a norm of Sobolev type provided that the flux collocation points are placed at the zeros of the corresponding Legendre polynomial [17]. The essential connections between FR and DG methods have been analyzed in [18], [19]. A series of flux reconstruction methods have been developed correspondingly [20], [21], [22], [23], [24]. Specifically, Vincent et al. proposed a new class of energy stable flux reconstruction methods based on Huynh's approach, which is often referred as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes [20]. In what follows, we refer the terminology Flux Reconstruction corresponding to Vincent's formulation if unspecified.
Another hot topic in computational fluid dynamics research might go into the study of multi-scale and non-equilibrium flow dynamics. As an example, the Boltzmann equation provides a statistical description of particle transports and collisions at the mesoscopic scale, i.e. the molecular mean free path and collision time. The evolution of one-particle probability density function is followed within the phase space. Compared to macroscopic fluid equations, the Boltzmann equation provides many more degrees of freedom and thus can be used to describe both equilibrium and non-equilibrium systems. Hilbert's 6th problem [25] served as an intriguing beginning of trying to link the behaviors of an interacting many-particle system across different scales. It has been shown since then that hydrodynamic equations can be recovered from the asymptotic limits of the Boltzmann solutions [26], [27], [28].
The challenge of solving the Boltzmann equation mainly comes from two sources. First, the equation is built upon a seven-dimensional phase space, which is nonlinearly coupled to depict particle transports. Second, the collision operator of the Boltzmann equation is a complicated fivefold integral with three from velocity space and two from a unit sphere. A common compromise in numerical simulations is to replace the full Boltzmann collision integral with relaxation terms [29]. The simplified equations thus obtained are similar as radiation and neutron transport equations, where continuous efforts have been devoted to the construction of high-order numerical solvers [30], [31], [32]. Boscheri and Dimarco [33] developed a class of central WENO implicit-explicit Runge Kutta schemes for the BGK model of the Boltzmann equation. Groppi et al. followed a semi-Lagrangian formulation of the BGK equation and employed diagonally implicit Runge Kutta and back differentiation formula to construct high-order schemes [34]. Xiong et al. [35] used nodal discontinuous Galerkin method and constructed asymptotic preserving schemes for the BGK equation in a hyperbolic scaling.
The earlier numerical solvers for Boltzmann collision integral were mostly based on the point-to-point principle [36], i.e., the post-collision particle velocities also need to fall onto the velocity grid. It was then proved that the computational cost of such methods is of , where N is the number of velocity grids in each direction, and the convergence order of accuracy is less than one [37]. Another idea goes to solve the collision term in spectral space by means of Fourier transforms. Bobylev [38] made a preliminary attempt to calculate the Boltzmann equation for Maxwell molecules in spatially uniform field. This method was then extended to general collision kernels with a computational cost [39]. In 2006, Mouhot and Pareschi proposed a fast spectral method based on the Carleman-type Boltzmann collision operator with the cost . Here M is the number of grid points for discretizing polar angles, which is much smaller than the number of velocity grids N in each direction. The advantageous efficiency enables the full Boltzmann simulation of multi-dimensional fluid dynamic problems [40], [41], [42], [43], [44], [45], [46].
The existing attempts on constructing high-order Boltzmann solvers are very limited, of which the following two are known to us. Jaiswal et al. [42] developed a discontinuous Galerkin fast spectral method in conjunction with Runge-Kutta integrator. Su et al. [43] built an implicit discontinuous Galerkin solver in the iterative fashion. In fact, given the complexity in evaluating the collision term, it points a promising direction to construct high-order methods for the full Boltzmann equation. Thanks to the higher accuracy achieved per computational degree of freedom, the use of high-order methods leads to a reduction of elements and can thereby reduce space and time complexity for numerical solution.
In this paper, a novel flux reconstruction kinetic scheme (FRKS) is presented for the Boltzmann equation. Based on the Lagrange interpolation and reconstruction, the kinetic upwind flux functions are solved simultaneously within physical and particle velocity space. The fast spectral method is incorporated into the FR framework to solve the full Boltzmann collision integral. The explicit singly diagonally implicit Runge-Kutta (ESDIRK) method [47] is incorporated as numerical integrator and thus the stiffness of the collision operator in the continuum flow regime can be overcome. We ensure the shock capturing property by introducing a self-adaptive artificial dissipation, which is derived from the effective cell Knudsen number at the kinetic scale. As a result, the FRKS is able to capture the cross-scale flow dynamics where resolved and unresolved regions coexist inside a flow field.
The rest of this paper is organized as follows. Section 2 is a brief introduction of the kinetic theory of gases. Section 3 presents the formulation of the solution algorithm and its detailed implementation. Section 4 includes numerical experiments to demonstrate the performance of the flux reconstruction kinetic scheme. The last section is the conclusion.
Section snippets
Kinetic theory
The gas kinetic theory describes the time-space evolution of particle distribution function . With a separate modeling of particle transport and collision processes, the Boltzmann equation of dilute monatomic gas in the absence of external force is where are the pre-collision velocities of two classes of colliding particles, and are the corresponding post-collision velocities. The collision kernel
Formulation
Considering the domain Ω with N non-overlapping cells we represent the solution of the Boltzmann equation with piecewise polynomials. Within each element , the particle distribution function is approximated by a polynomial of degree m denoted , and the corresponding flux function is approximated of degree , i.e. . Therefore, the total approximate solutions are
For convenience, a standard coordinate can be introduced
Numerical experiments
In this section, we will conduct numerical experiments to validate the current scheme. In order to demonstrate the cross-scale computing capability of the algorithm, the results at different degrees of gas rarefaction are presented. The variables are non-dimensionalized in the same way as in section 3.5. The monatomic gas is considered in all cases.
Conclusion
Non-equilibrium statistical mechanics is profoundly built upon the Boltzmann equation. For the first time, a high-order kinetic scheme based on flux reconstruction is proposed for solving the Boltzmann equation in this paper. The upwind flux solver is integrated with flux reconstruction formulation seamlessly throughout the phase space. The fast spectral method is constructed to solve the exact Boltzmann collision integral with an arbitrary collision kernel. Besides, the explicit singly
CRediT authorship contribution statement
Tianbai Xiao: Conceptualization, Formal analysis, Investigation, Methodology, Project administration, Resources, Software, Visualization, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The author would like to acknowledge the fruitful discussion with Jiaqing Kou. The current research is funded by the Alexander von Humboldt Foundation (Ref3.5-CHN-1210132-HFST-P).
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