A DGBGK scheme based on WENO limiters for viscous and inviscid flows

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Abstract

This paper presents a discontinuous Galerkin BGK (DGBGK) method for both viscous and inviscid flow simulations under a DG framework with a gas-kinetic flux and WENO limiters. In the DGBGK method, the construction of the flux in the DG method is based on the particle transport and collisional mechanism which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation. Due to the connection between the gas-kinetic BGK model and the Euler as well as the Navier–Stokes equations, both viscous and inviscid flow equations can be simulated by a unified formulation. WENO limiters are used to obtain uniform high-order accuracy and sharp non-oscillatory shock transition. In the current method, the time accuracy is achieved by the direct integration of both time-dependent flux function at a cell interface and the flow variables inside each element. Numerical examples in one and two space dimensions are presented to illustrate the robustness and accuracy of the present scheme.

Introduction

In hydrodynamic simulations, the finite volume (FV) and the discontinuous Galerkin (DG) finite element methods have been successfully developed and used in a wide range of applications. Most FV schemes use piecewise constant representation of flow variables inside each control volume and employ the reconstruction techniques to obtain high accuracy. Since a higher-order scheme usually uses a wider stencil than that in a lower-order scheme, difficulties emerge in its implementation in the flow computation on unstructured meshes with complex geometry. However, for the DG method, high-order accuracy is achieved by means of high-order polynomial approximation within each element rather than by means of wide stencils, where more information is stored and updated for each element in the computation. Because only neighboring elements interaction is included, the DG method becomes much easy and efficient in its application on unstructured meshes. The easy handling of boundary condition is another benefit. Moreover, the use of discontinuous polynomial approximation produces a block diagonal mass matrix which becomes efficient to deal with numerically. At the same time, the slope limiting techniques can be incorporated in the DG method in a natural way. Explicit Runge–Kutta method is usually used for the time discretization in the DG method, which makes the resulting algorithm highly parallel. Now, the DG method has served as a high-order method for a broad class of problems, see, for example [6], [9], [7], [8], [10]. There is an extensive literature devoting to the study of the DG method for viscous and inviscid flows, such as [1], [2], [4], [27], [11].

For flows with strong discontinuities, the direct update of flow variables in the DG framework generates numerical oscillations. In order to get an oscillation-free solution, the limiting techniques used in the shock capturing upwind schemes were adopted here. In [23], the WENO limiters are successfully used in the DG method, where the main idea is to abandon the polynomial solution in the “troubled” cells and to reconstruct new polynomials with the information from neighboring cells. The use of the limiting techniques is the main reason for the success to capture shock discontinuities [16], [12], [14], [20].

The gas-kinetic BGK scheme, proposed by Prendergast and Xu [22], [28], [29], is a finite volume method which makes use of the local integral solution of the collisional BGK model to compute a time-dependent gas distribution function at a cell interface and to obtain the numerical fluxes in the gas evolution stage. Since the BGK model is a statistical model, the particle transport and collision are coupled in the whole gas evolution process, and the particle collision time controls the physical dissipative coefficients in the macroscopic equations. Since the gas evolution is associated with a relaxation process, i.e., from a non-equilibrium state to an equilibrium one, the entropy condition is always satisfied by the BGK scheme. Based on the Chapman–Enskog expansion, from the gas-kinetic BGK model the Euler as well as the Navier–Stokes equations can be derived. In the smooth flow region, as the flow structure can be well resolved by the numerical cell size, the BGK scheme goes back to the Lax–Wendroff-type method for the compressible Navier–Stokes equations. In the discontinuity region, a delicate dissipative mechanism due to both kinematic and dynamic dissipation in the BGK scheme presents a stable and crisp shock transition, see [30], [21]. Many engineering flow problems have been studied using the BGK scheme [17], [18]. Another advantage to use the kinetic approach in the flux evaluation is due to the fact that the flux for the higher-order equations, such as Burnett and Super-Burnett, can be easily constructed [31]. Also, the physical modeling, such as gravity accelerating the particle movement or multicomponent gas interaction, can be easily implemented in the gas-kinetic formulation to design physically reliable schemes.

Hybrid schemes which inherit the merits of both the DG method and gas-kinetic schemes have been recently investigated by some authors. Based on the use of the collisionless Boltzmann equation, Tang and Warnecke proposed a gas-kinetic RKDG method for inviscid flows [25], where the accuracy and efficiency have been numerically demonstrated. Xu has proposed a DG-based BGK scheme for viscous fluids [32] for which a lower time integration method for the flow variables inside each cell was used. Recently, a Runge–Kutta DG–BGK scheme was presented for viscous fluids in [19] where a Runge–Kutta time discretization was used. For the schemes in [32], [19] the usual TVD limiters have been employed.

In this paper, we shall incorporate the DG method with the BGK scheme to propose a DGBGK method for both inviscid and viscous flow simulations. In comparison with previous work, for the first time the following two recipes are presented. On the one hand, high-order time accuracy is obtained directly using a time-dependent flux function at a cell interface instead of implementing the Runge–Kutta or TVD–RK time discretization in the DG method [23], [11]. Previously, a Runge–Kutta DG method incorporating with the BGK scheme was proposed in [19], and a lower-order time evolution method was used for the time integration inside each element in [32]. On the other hand, in the current paper the WENO limiters are successfully used in the DGBGK scheme, where both [32], [19] used usual TVD limiter in the corresponding DG methods for viscous fluids.

The paper is organized as follows. Section 2 describes the DGBGK method. For the sake of simplicity, the detail algorithm is given only for one-dimensional formulation, and its extension to two-dimensional space is discussed briefly in Section 2.6. More specifically, Section 2.1 describes the relation between the BGK model and the compressible Euler as well as Navier–Stokes equations. Section 2.2 presents the formulation for the DG method. A brief description of construction of the BGK fluxes is given in Section 2.3. In Sections 2.4 Time discretization, 2.5 WENO limiter, the time discretization and the limiting procedure are presented. The performance of the proposed method is illustrated in Section 3 through many numerical examples in both 1D and 2D cases. Finally, a brief comparison of the computational cost between the DGBGK scheme and the usual DG method with WENO limiters is presented.

Section snippets

A DGBGK scheme

Similar to many other finite volume methods, the gas-kinetic scheme is mainly about the flux evaluation at cell interfaces. The distinguishable feature of the gas-kinetic BGK scheme is that a Navier–Stokes flux is given directly from the MUSCL-type reconstructed initial data [15]. In the DG–WENO method, a high-order polynomial solution within each element is updated by using DG formulation and limited by using WENO limiters. In this section, we shall present a DGBGK scheme by incorporating the

Numerical experiments

In order to validate the DGBGK scheme, we will present five numerical examples in this section. Three of them are the one-dimensional simulations and other two are two-dimensional flows.

Conclusion

A DGBGK scheme has been developed for both viscous and inviscid flow simulations using a DG framework with a gas-kinetic flux and WENO limiters. The new scheme inherits some merits of both the DG and the BGK methods. The construction of the flux in the DGBGK scheme is based on the particle transport and collisional mechanism which not only couples the convective and dissipative terms, but also includes both discontinuous and continuous flow distributions in the flux evaluation at cell

Acknowledgement

This work was supported by the National Basic Research Program (No. 2005CB321700), NSFC (Grant No. 10225105), and by a grant of Chinese Academy of Engineering Physics (No. 20060644). K. Xu was supported by Hong Kong Research Grant Council through RGC HKUST621005 and 621406.

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