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Register a new userDiscrete unified gas kinetic scheme for nonlinear convection-diffusion equations
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In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for general nonlinear convection-diffusion equation (NCDE), and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and find that the numerical results are in good agreement with analytical solutions and the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, DUGKS can also adopt the non-uniform mesh. Besides, we performed some comparisons among the DUGKS, finite-volume lattice Boltzmann model (FV-LBM), single-relaxation-time lattice Boltzmann model (SLBM) and multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the DUGKS model is more accurate than FV-LBM, more stable than SLBM, and almost has the same accuracy as the MRT-LBM. Besides, the using of non-uniform mesh may make DUGKS model more flexible.
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The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows which combines the benefits of both Lattice Boltzmann Method (LBM) and unified gas kinetic scheme (UGKS). By reconstruction of gas distribution function using particle velocity characteristic line, flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with reconstruction stage on a whole time step instead of half time step in original DUGKS. Using temporal/spatial integral Boltzmann Bhatnagar-Gross-Krook (BGK) equation, the transformed distribution function with inclusion of collision effect is constructed. The macro and mesoscopic fluxes of the cell on next time step is predicted by reconstruction of transformed distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on next time step can be updated through its macroscopic flux. Equilibrium distribution function on next time step can also be updated. Gas distribution function is updated by FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. Numerical time step is only limited by the Courant-Friedrichs-Lewy (CFL) condition and relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, as well as micro cavity flow cases are conducted to validate present scheme. The numerical simulation results agree well with the references results.
The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for deterministic solution of the Boltzmann model equation with asymptotic preserving property. In DUGKS, the numerical flux of the distribution function is determined from a local numerical solution of the Boltzmann model equation using an unsplitting approach. The time step and mesh resolution are not restricted by the molecular collision time and mean free path. To demonstrate the capacity of DUGKS in practical problems, this paper extends the DUGKS to arbitrary unstructured meshes. Several tests of both internal and external flows are performed, which include the cavity flow ranging from continuum to free molecular regimes, a multiscale flow between two connected cavities with a pressure ratio of 10000, and a high speed flow over a cylinder in slip and transitional regimes. The numerical results demonstrate the effectiveness of the DUGKS in simulating multiscale flow problems.
In this paper, an efficient high-order gas-kinetic scheme (EHGKS) is proposed to solve the Euler equations for compressible flows. We re-investigate the underlying mechanism of the high-order gas-kinetic scheme (HGKS) and find a new strategy to improve its efficiency. The main idea of the new scheme contains two parts. Firstly, inspired by the state-of-art simplifications on the third-order HGKS, we extend the HGKS to the case of arbitrary high-order accuracy and eliminate its unnecessary high-order dissipation terms. Secondly, instead of computing the derivatives of particle distribution function and their complex moments, we introduce a Lax-Wendroff procedure to compute the high-order derivatives of macroscopic quantities directly. The new scheme takes advantage of both HGKS and the Lax-Wendroff procedure, so that it can be easily extended to the case of arbitrary high-order accuracy with practical significance. Typical numerical tests are carried out by EHGKS, with the third, fifth and seventh-order accuracy. The presence of good resolution on the discontinuities and flow details, together with the optimal CFL numbers, validates the high accuracy and strong robustness of EHGKS. To compare the efficiency, we present the results computed by the EHGKS, the original HGKS and Runge-Kutta-WENO-GKS. This further demonstrates the advantages of EHGKS.
With a noticeable increase in research centered on modeling micro fluid interfaces in the framework of mesoscopic methods, we conduct an exhaustive study of discrete unified gas-kinetics scheme (DUGKS) in handling complicated interface deformations. High-order isotropic finite-difference schemes are first utilized in DUGKS to improve its capability in tracking interfaces. The performance of third-stage third-order DUGKS where source term is incorporated has also been assessed for the first time and a series of numerical tests have been conducted to investigate their capability. The comparative analysis have revealed the reason why the performance of lattice Boltzmann method is superior to that of discrete velocity method and DUGKS in general condition from an informed perspective. The mechanism behind the performance distinction between the central scheme and upwind scheme utilized in meso-flux construction in DUGKS have also been clarified. Numerical results have shown that the employment of high-order schemes in DUGKS does have an effect on the reduction of numerical dissipation, but the overall accuracy of this method is limited by the precision of prediction of source terms on mesh interface. The capability of third-stage third-order DUGKS is severely inhibited by its intrinsic limitation of the ratio of time step to particle collision time. Among the various kinds of DUGKS employed with different reconstruction methods, the most promising scheme is the one with third-order isotropic reconstruction and upwind-based meso-flux evaluation, which is able to ensure an unique balance between efficiency and accuracy.
In this paper, we perform a more general analysis on the discrete effects of some boundary schemes of the popular one- to three-dimensional DnQq multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation (CDE). Investigated boundary schemes include anti-bounce-back(ABB) boundary scheme, bounce-back(BB) boundary scheme and non-equilibrium extrapolation(NEE) boundary scheme. In the analysis, we adopt a transform matrix $textbf{M}$ constructed by natural moments in the evolution equation, and the result of ABB boundary scheme is consistent with the existing work of orthogonal matrix $textbf{M}$. We also find that the discrete effect does not rely on the choice of transform matrix, and obtain a relation to determine some of the relaxation-time parameters which can be used to eliminate the numerical slip completely under some assumptions. In this relation, the weight coefficient is considered as an adjustable parameter which makes the parameter adjustment more flexible. The relaxation factors associated with second moments can be used to eliminate the numerical slip of ABB boundary scheme and BB boundary scheme while the numerical slip can not be eliminated of NEE boundary scheme. Furthermore, we extend the relations to complex-valued CDE, several numerical examples are used to test the relations.
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