Microscopically implicit–macroscopically explicit schemes for the BGK equation

In this work a new class of numerical methods for the BGK model of kinetic equations is introduced. The schemes proposed are implicit with respect to the distribution function, while the macroscopic moments are evolved explicitly. In this fashion, the stability condition on the time step coincides with a macroscopic CFL, evaluated using estimated values for the macroscopic velocity and sound speed. Thus the stability restriction does not depend on the relaxation time and it does not depend on the microscopic velocity of energetic particles either. With the technique proposed here, the updating of the distribution function requires the solution of a linear system of equations, even though the BGK model is highly non linear. Thus the proposed schemes are particularly effective for high or moderate Mach numbers, where the macroscopic CFL condition is comparable to accuracy requirements. We show results for schemes of order 1 and 2, and the generalization to higher order is sketched.     

Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.     

Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium

Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (“kinetic”) relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as “unphysical”. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.     

On large time step TVD scheme for hyperbolic conservation laws and its efficiency evaluation

A large time step (LTS) TVD scheme originally proposed by Harten is modified and further developed in the present paper and applied to Euler equations in multidimensional problems. By firstly revealing the drawbacks of Harten’s original LTS TVD scheme, and reasoning the occurrence of the spurious oscillations, a modified formulation of its characteristic transformation is proposed and a high resolution, strongly robust LTS TVD scheme is formulated. The modified scheme is proven to be capable of taking larger number of time steps than the original one. Following the modified strategy, the LTS TVD schemes for Yee’s upwind TVD scheme and Yee–Roe–Davis’s symmetric TVD scheme are constructed. The family of the LTS schemes is then extended to multidimensional by time splitting procedure, and the associated boundary condition treatment suitable for the LTS scheme is also imposed. The numerical experiments on Sod’s shock tube problem, inviscid flows over NACA0012 airfoil and ONERA M6 wing are performed to validate the developed schemes. Computational efficiencies for the respective schemes under different CFL numbers are also evaluated and compared. The results reveal that the improvement is sizable as compared to the respective single time step schemes, especially for the CFL number ranging from 1.0 to 4.0.     

Investigation of Stability and Hydrodynamics of Different Lattice Boltzmann Models

Stability and hydrodynamic behaviors of different lattice Boltzmann models including the lattice Boltzmann equation (LBE), the differential lattice Boltzmann equation (DLBE), the interpolation-supplemented lattice Boltzmann method (ISLBM) and the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) are studied in detail. Our work is based on the von Neumann linearized stability analysis under a uniform flow condition. The local stability and hydrodynamic (dissipation) behaviors are studied by solving the evolution operator of the linearized lattice Boltzmann equations numerically. Our investigation shows that the LBE schemes with interpolations, such as DLBE, ISLBM and TLLBM, improve the numerical stability by increasing hyper-viscosities at large wave numbers (small scales). It was found that these interpolated LBE schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper-viscosities.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

Stability analysis of a predictor/multi-corrector method for staggered-grid Lagrangian shock hydrodynamics

This article presents the complete von Neumann stability analysis of a predictor/multi-corrector scheme derived from an implicit mid-point time integrator often used in shock hydrodynamics computations in combination with staggered spatial discretizations. It is shown that only even iterates of the method yield stable computations, while the odd iterates are, in the most general case, unconditionally unstable. These findings are confirmed by, and illustrated with, a number of numerical computations. Dispersion error analysis is also presented.     

An adaptive central-upwind weighted essentially non-oscillatory scheme

In this work, an adaptive central-upwind 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. The scheme adapts between central and upwind schemes smoothly by a new weighting relation based on blending the smoothness indicators of the optimal higher order stencil and the lower order upwind stencils. The scheme achieves 6th-order accuracy in smooth regions of the solution by introducing a new reference smoothness indicator. A number of numerical examples suggest that the present scheme, while preserving the good shock-capturing properties of the classical WENO schemes, achieves very small numerical dissipation.     

Modified kinetic flux vector splitting schemes for compressible flows

We investigate the traditional kinetic flux vector splitting (KFVS) and BGK schemes for the compressible Euler equations. First, based on a careful study of the behavior of the discrete physical variables across the contact discontinuity, we analyze quantitatively the mechanism of inducing spurious oscillations of the velocity and pressure in the vicinity of the contact discontinuity for the first-order KFVS and BGK schemes. Then, with the help of this analysis, we propose a first-order modified KFVS (MKFVS) scheme which is oscillation-free in the vicinity of the contact discontinuity, provided certain consistent conditions are satisfied. Moreover, by using piecewise linear reconstruction and van Leer’s limiter, the first-order MKFVS scheme is extended to a second-order one, consequently, a nonoscillatory second-order MKFVS scheme is constructed. Finally, by combing the MKFVS schemes with the γ-model, we successfully extend the MKFVS schemes to multi-flows, and propose therefore a first- and second-order MKFVS schemes for multi-fluid computations, which are nonoscillatory across fluid interfaces. A number of numerical examples presented in this paper validate the theoretic analysis and demonstrate the good performance of the MKFVS schemes in simulation of contact discontinuities for both single- and multi-fluids.     

CFL Conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We study time step restrictions due to linear stability constraints of Runge–Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax–Friedrichs flux or the upwind flux, and integrated in time with various Runge–Kutta schemes designed for linear wave propagation problems or non-linear applications. Von–Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order p ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax–Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.     

QUICK与多种差分方案的比较和计算

本文用QUICK和多种差分方案计算了四个流动与换热问题．计算结果表明。对于强制流动问题,QUICK用较粗网格就能得到其他差分方案用较细网格才能得到的结果。对稳态自然对流,QUICK与其他差分方案的计算结果相近,但QUICK方案能预测出所计算的低Pr数流体自然对流的物理振荡,而其他几种方案不能．     

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

Is the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

A Remark on the Courant-Friedrichs-Lewy Condition in Finite Difference Approach to PDE's

The Courant-Friedrichs-Lewy condition (The CFL condition) is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations. We give a remark on the CFL condition from a view point of stability, and we give some numerical experiments which show instability of numerical solutions even under the CFL condition. We give a mathematical model for rounding errors in order to explain the instability.     

非结构网格上的迎风有限元格式

在非结构网格上提出一种基于修正积分区域的迎风有限元格式,它与一阶迎风差分格式相当,可应用于构造各种不同的数值格式。     

Finite difference lattice Boltzmann method with arbitrary specific heat ratio applicable to supersonic flow simulations

The current finite difference lattice Boltzmann method (FDLBM) gives a fixed specific heat ratio because internal energy is limited to the translational freedom of the space. Yan et al. and Kataoka et al. clarified the conditions for deriving models with arbitrary specific heat ratio and proposed Euler models. However, these model applications to numerical simulations showed the weakness in the numerical stability. In this paper, a two-dimensional FDLBM Navier Stokes model and a three-dimensional FDLBM Euler model, which allow arbitrary values to be set for the specific heat ratio, were proposed. These models stably performed numerical simulations from subsonic to supersonic ranges.     

Basic function method A new numerical method on unstructured grids

A new numerical method-basic function method is proposed. This method can directly discrete differential operators on unstructured grids. By using the expansion of basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the polynomial as basic function, applying the technique of flux splitting method and the combination of central and upwind schemes, the non-physical fluctuation near the shock wave is suppressed. The first-order basic function scheme of polynomial type for solving inviscid compressible flow numerically is constructed in this paper. Several numerical results of many typical examples for one-, two- and three-dimensional inviscid compressible steady flow illustrate that it is a new scheme with high accuracy and high resolution for shock wave. Especially, combining with the adaptive remeshing technique, the satisfactory results can be obtained by these schemes.     

Time Step Validation Method Research for Low-Prandtl Number Fluid Numerical Simulation

The stability condition of Courant number and diffusion number is proved for an SGSD (stability guaranteed second-order difference) scheme by von Neumann method in implicit and explicit discretization of the one-dimensional convection and diffusion terms. Then, a series of numerical simulations of fluid flow and heat transfer based on two-dimensional unsteady state model is used to study the combined natural and MHD (magnetohydrodynamics) convection in a Joule-heated cavity using the finite volume methods, for the fluid of Pr = 0.01, also we use an SGSD scheme and IDEAL (inner doubly iterative efficient algorithm for linked equations) algorithm. It is found that periodic oscillation flow evolves.We propose a new convergence concept for the simulation oscillation results; the results of the numerical experiments are presented and they confirm our theoretical conclusions. The convergence result is checked in another way. It is found that the two approaches have the same results and can judge the validity of the time step. The proposed method is helpful to get reliable results efficiently.     

Stability Analysis of a Fully Coupled Implicit Scheme for Inviscid Chemical Non-Equilibrium Flows

Von Neumann stability theory is applied to analyze the stability of a fully coupled implicit (FCI) scheme based on the lower-upper symmetric Gauss-Seidel (LU-SGS) method for inviscid chemical non-equilibrium flows. The FCI scheme shows excellent stability except the case of the flows involving strong recombination reactions, and can weaken or even eliminate the instability resulting from the stiffness problem, which occurs in the subsonic high-temperature region of the hypersonic flow field. In addition, when the full Jacobian of chemical source term is diagonalized, the stability of the FCI scheme relies heavily on the flow conditions. Especially in the case of high temperature and subsonic state, the CFL number satisfying the stability is very small. Moreover, we also consider the effect of the space step, and demonstrate that the stability of the FCI scheme with the diagonalized Jacobian can be improved by reducing the space step. Therefore, we propose an improved method on the grid distribution according to the flow conditions. Numerical tests validate sufficiently the foregoing analyses. Based on the improved grid, the CFL number can be quickly ramped up to large values for convergence acceleration.