## 1 Introduction
By truncating the Chapman-Enskog (CE) expansion into different orders, the approach leads to respectively the Euler, NS, Burnett, and super-Burnett equations at the zeroth-, first-, second-, and third-order approximations [30]. However, [PERSON] and [PERSON] have pointed out that the Burnett equations violate the underlying physics behind the Boltzmann equation [31], and [PERSON] has argued that as the order of Chapman-Enskog expansion increases, the accuracy of the obtained macroscopic equations does not necessary increase [32]. In 1949, [PERSON] proposed a new way to derive the macroscopic equations via the method of moment [33; 34]. Unlike the CE expansion in which the molecular VDF is expanded in powers of the Knudsen number, [PERSON] expanded the molecular VDF in Hermite polynomials [35]. In addition to the mass, momentum, and energy conservation laws, the governing equations for the shear stress and heat flux were obtained by multiplying the moment of the peculiar velocity \(c_{i}=C_{i}-u_{i}\) with the Boltzmann equation and integrating over the velocity space \(C_{i}\). This set of equations is called G13 moment equations, which can be closed by expanding the VDF into the Hermite polynomials [23; 24]. [PERSON] and [PERSON] [36; 37] and [PERSON] [27] regularized the G13 moment equations (denoted as the R13 moment equations) by applying a CE-like expansion and an order-of magnitude approach. [PERSON] and [PERSON] [38] and [PERSON] and [PERSON] [37; 39] obtained the wall boundary conditions (WBCs) for the regularized 13 moment equations (R13) based on the Maxwell's kinetic WBC [40]. The capability of the R13 moment equations in modeling microscale gas phenomena have been investigated [41], and the R13 moment method can capture the non-equilibrium effects with the Knudsen number up to 0.25 [41; 42]. However, they cannot provide sufficient accurate description of the Knudsen layer [43; 44]. To remedy the deficiency of the R13 equations, [PERSON] and [PERSON] extended the theory of moment to regularized 26 (R26) equations [42], which is demonstrated as a potential engineering design tool for the non-equilibrium flows in the early transition regime [45; 46; 47; 48; 49]. The R26 moment equations have been successfully applied in the study of pressure driven Poiseuille flow [42; 45], thermal transpiration flow [46], gas flows in porous media [47; 48] and non-equilibriumrarefied flow past a circular cylinder [49]. However, a detailed comparative study between the moment method and the DVM, in terms of the accuracy and efficiency, at both the microscopic and macroscopic levels, remains relatively scarce in the literature. Therefore, the purpose of this paper is to give a detailed comparison of the moment methods and the discrete velocity method and provide essential information for users to choose the appropriate one for applications. The remaining part of this paper is organized as follows. We first make an overview of the Boltzmann model equation and the discrete velocity method, as described in Section 2. The method of moments, including the R13 and R26 moment methods, are introduced briefly in Section 3, followed by the evaluation of the reconstructed VDFs, as described in Section 4. Numerical simulations of several types of typical non-equilibrium flows (1D force-driven Poiseuille flow, 2D lid-driven cavity flow, and thermal cavity flow induced by temperature gradients) are presented and discussed in Section 5 in comparison with the DVM data. A brief summary is finally given in Section 6.