Consistency with continuity in conservative advection schemes for free‐surface models

The consistency of the discretization of the scalar advection equation with the discretization of the continuity equation is studied for conservative advection schemes coupled to three‐dimensional flows with a free‐surface. Consistency between the discretized free‐surface equation and the discretized scalar transport equation is shown to be necessary for preservation of constants. In addition, this property is shown to hold for a general formulation of conservative schemes. A discrete maximum principle is proven for consistent upwind schemes. Various numerical examples in idealized and realistic test cases demonstrate the practical importance of the consistency with the discretization of the continuity equation. Copyright © 2002 John Wiley & Sons, Ltd.     

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

A gas-kinetic numerical method for directly solving the mesoscopic velocity distribution function equation is presented and applied to the study of three-dimensional complex flows and micro-channel flows covering various flow regimes. The unified velocity distribution function equation describing gas transport phenomena from rarefied transition to continuum flow regimes can be presented on the basis of the kinetic Boltzmann–Shakhov model equation. The gas-kinetic finite-difference schemes for the velocity distribution function are constructed by developing a discrete velocity ordinate method of gas kinetic theory and an unsteady time-splitting technique from computational fluid dynamics. Gas-kinetic boundary conditions and numerical modeling can be established by directly manipulating on the mesoscopic velocity distribution function. A new Gauss-type discrete velocity numerical integration method can be developed and adopted to attack complex flows with different Mach numbers. HPF parallel strategy suitable for the gas-kinetic numerical method is investigated and adopted to solve three-dimensional complex problems. High Mach number flows around three-dimensional bodies are computed preliminarily with massive scale parallel. It is noteworthy and of practical importance that the HPF parallel algorithm for solving three-dimensional complex problems can be effectively developed to cover various flow regimes. On the other hand, the gas-kinetic numerical method is extended and used to study micro-channel gas flows including the classical Couette flow, the Poiseuille- channel flow and pressure-driven gas flows in two-dimensional short micro-channels. The numerical experience shows that the gas-kinetic algorithm may be a powerful tool in the numerical simulation of micro-scale gas flows occuring in the Micro-Electro-Mechanical System (MEMS). The project supported by the National Natural Science Foundation of China (90205009 and 10321002), and the National Parallel Computing Center in Beijing. The English text was polished by Yunming Chen.     

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.     

不可压N-S方程高效算法及二维槽道湍流分析

构造了基于非等距网格的迎风紧致格式,并将其与三阶精度的Adams半隐方法相结合,构造了求解不可压N－S方程高效算法。该算法利用基于交错网格的离散形式的压力Poisson方程求解压力项,解决了边界处的残余散度问题;同时还利用快速Fourier变换将方程的隐式部分解耦,离散后的代数方程组利用追赶法求解,大大减少了计算量。通过对二维槽道流动的数值模拟,证实了所构造的数值方法具有精度高,稳定性好,能抑制混淆误差等优点,同时具有很高的计算效率,是进行壁湍流直接数值模拟的有效方法。在数值模拟的基础上对二维槽道流动进行了分析,得到了Reynolds数从6000到15000的二维流动饱和态解（所谓“二维槽道湍流”）;定性及定量结果均与他人的数值计算结果吻合十分理想。对流场进行了分析,指出了“二维湍流”与三维湍流统计特性的区别。     

Accurate and efficient simulation of transport in multidimensional flow

A new numerical method to obtain high‐order approximations of the solution of the linear advection equation in multidimensional problems is presented. The proposed conservative formulation is explicit and based on a single updating step. Piecewise polynomial spatial discretization using Legendre polynomials provides the required spatial accuracy. The updating scheme is built from the functional approximation of the exact solution of the advection equation and a direct evaluation of the resulting integrals. The numerical details for the schemes in one and two spatial dimensions are provided and validated using a set of numerical experiments. Test cases have been oriented to the convergence and the computational efficiency analysis of the schemes. Copyright © 2009 John Wiley & Sons, Ltd.     

A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations

In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy.     

Charge-conserving grid based methods for the Vlasov–Maxwell equations

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver.     

Numerical boundary conditions for globally mass conservative methods to solve the shallow‐water equations and applied to river flow

A revision of some well‐known discretization techniques for the numerical boundary conditions in 1D shallow‐water flow models is presented. More recent options are also considered in the search for a fully conservative technique that is able to preserve the good properties of a conservative scheme used for the interior points. Two conservative numerical schemes are used as representatives of the families of explicit and implicit numerical methods. The implementation of the different boundary options to these schemes is compared by means of the simulation of several test cases with exact solution. The schemes with the global conservation boundary discretization are applied to the simulation of a real river flood wave leading to very satisfactory results. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

In the conventional discrete velocity method (DVM), the local solution of collisionless Boltzmann equation with a piecewise constant distribution for the distribution function is utilized to reconstruct distribution function at the cell interface and then calculate numerical flux of Boltzmann equation for updating the distribution function at cell center. In this process, a numerical dissipation will be introduced into the solution due to neglecting of the collision effect at the cell interface. This numerical dissipation may deteriorate the solution accuracy of conventional DVM in the continuum flow regime, in which the particle collision happens frequently. To overcome this defect, two improved schemes are first presented in this work, in which the local discrete solution of Boltzmann equation with Shakhov model is adopted to evaluate the distribution function at the cell interface, while the equilibrium state of the local solution is computed by different ways. One of the improved schemes evaluates the equilibrium state exactly by the moments of distribution functions according to the compatibility condition, while the other computes the equilibrium state approximately by a simple average at the cell interface. Since the collision effect is incorporated in evaluation of numerical flux, the improved schemes can provide reasonable solutions in all flow regimes. On the other hand, they introduce some extra computational efforts for determining the collision term at the cell interface as compared with the conventional DVM. To assess the performance of different methods for simulation of flows in all flow regimes, a comprehensive study is then carried out in this work.     

两个新型的八结点块体单元—三维离散算子差分法

给出了弹性力学三维问题的离散算子差分法 ,讨论离散算子差分法在三维问题中的特点 ,意在为该方法的进一步发展提供依据 ,为应用弱形式进行数值求解的研究提供参考。本文从弹性力学平衡方程更为一般的弱形式出发 ,给出了含边界参数的弱形式方程。由该方程不仅可以得到有限元法 ,还可得到离散算子差分法。给出了两个八结点块体单元 ,虽然单元中位移函数是非协调的 ,不需特殊处理便可保证离散格式收敛 ,并对单元位移有十分好的反映能力。     

Positive‐definite q‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

A new family of locally conservative cell‐centred flux‐continuous schemes is presented for solving the porous media general‐tensor pressure equation. A general geometry‐permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control‐volume distributed subcell flux‐continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2 :259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive‐definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical‐space schemes are shown to be non‐symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non‐symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical‐space and subcell‐space q‐families of schemes. M‐matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical‐space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell‐wise constant tensor schemes and that subcell tensor approximation using the control‐volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd.     

Consistent discretization for simulations of flows with moving generalized curvilinear coordinates

We develop a consistent discretization of conservative momentum and scalar transport for the numerical simulation of flow using a generalized moving curvilinear coordinate system. The formulation guarantees consistency between the discrete transport equation and the discrete mass conservation equation due to grid motion. This enables simulation of conservative transport using generalized curvilinear grids that move arbitrarily in three dimensions while maintaining the desired properties of the discrete transport equation on a stationary grid, such as constancy, conservation, and monotonicity. In addition to guaranteeing consistency for momentum and scalar transport, the formulation ensures geometric conservation and maintains the desired high‐order time accuracy of the discretization on a moving grid. Through numerical examples we show that, when the computation is carried out on a moving grid, consistency between the discretized scalar advection equation and the discretized equation for flow mass conservation due to grid motion is required in order to obtain stable and accurate results. We also demonstrate that significant errors can result when non‐consistent discretizations are employed. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite volume solver for 1D shallow‐water equations applied to an actual river

This paper describes the numerical solution of the 1D shallow‐water equations by a finite volume scheme based on the Roe solver. In the first part, the 1D shallow‐water equations are presented. These equations model the free‐surface flows in a river. This set of equations is widely used for applications: dam‐break waves, reservoir emptying, flooding, etc. The main feature of these equations is the presence of a non‐conservative term in the momentum equation in the case of an actual river. In order to apply schemes well adapted to conservative equations, this term is split in two terms: a conservative one which is kept on the left‐hand side of the equation of momentum and the non‐conservative part is introduced as a source term on the right‐hand side. In the second section, we describe the scheme based on a Roe Solver for the homogeneous problem. Next, the numerical treatment of the source term which is the essential point of the numerical modelisation is described. The source term is split in two components: one is upwinded and the other is treated according to a centred discretization. By using this method for the discretization of the source term, one gets the right behaviour for steady flow. Finally, in the last part, the problem of validation is tackled. Most of the numerical tests have been defined for a working group about dam‐break wave simulation. A real dam‐break wave simulation will be shown. Copyright © 2002 John Wiley & Sons, Ltd.     

气体动理学格式研究进展

介绍了近年来气体动理学格式(gas-kinetic scheme, GKS, 亦简称BGK 格式) 的主要研究进展, 重点是高阶精度动理学格式及适合从连续流到稀薄流全流域的统一动理学格式. 通过对速度分布函数的高阶展开和对初值的高阶重构, 构造了时间和空间均为三阶精度的气体动理学格式. 研究表明, 相比于传统的基于Riemann 解的高阶格式, 新格式不仅考虑了网格单元界面上物理量的高阶重构, 而且在初始场的演化阶段耦合了流体的对流和黏性扩散, 也能够保证解的高阶精度. 该研究为高精度计算流体力学(computatial uiddymamics, CFD) 格式的建立提供了一条新的途径. 通过分子离散速度空间直接求解Boltzmann 模型方程,在每个时间步长内将宏观量的更新和微观气体分布函数的更新紧密地耦合在一起, 建立了适合任意Knudsen(kn) 数的统一格式, 相比于已有的直接离散格式具有更高的求解效率. 最后, 本文还讨论了合理的物理模型对数值方法的重要性. 气体动理学方法的良好性能来自于Boltzmann 模型方程对计算网格单元界面上初始间断的时间演化的准确描述. 气体自由运动与碰撞过程的耦合是十分必要的. 通过分析数值激波层内的耗散机制,我们认识到采用Euler 方程的精确Riemann 解作为现代可压缩CFD 方法的基础具有根本的缺陷, 高马赫数下的激波失稳现象不可避免. 气体动理学格式为构造数值激波结构提供了一个重要的可供参考的物理机制.      

PRICE: primitive centred schemes for hyperbolic systems

We present first‐ and higher‐order non‐oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non‐conservative) form. Non‐conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non‐conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second‐order non‐oscillatory methods. We have used the MUSCL‐Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher‐order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non‐conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind‐based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd.     

Gas-kinetic unified algorithm for plane external force-driven flows covering all flow regimes by modeling of Boltzmann equation

The nonequilibrium steady gas flows under the external forces are essentially associated with some extremely complicated nonlinear dynamics, due to the acceleration or deceleration effects of the external forces on the gas molecules by the velocity distribution function. In this article, the gas-kinetic unified algorithm (GKUA) for rarefied transition to continuum flows under external forces is developed by solving the unified Boltzmann model equation. The computable modeling of the Boltzmann equation with the external force terms is presented at the first time by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The conservative discrete velocity ordinate (DVO) method is utilized to transform the governing equation into the hyperbolic conservation forms at each of the DVO points. The corresponding numerical schemes are constructed, especially the forward-backward MacCormack predictor-corrector method for the convection term in the molecular velocity space, which is unlike the original type. Some typical numerical examples are conducted to test the present new algorithm. The results obtained by the relevant direct simulation Monte Carlo method, Euler/Navier-Stokes solver, unified gas-kinetic scheme, and moment methods are compared with the numerical analysis solutions of the present GKUA, which are in good agreement, demonstrating the high accuracy of the present algorithm. Besides, some anomalous features in these flows are observed and analyzed in detail. The numerical experience indicates that the present GKUA can provide potential applications for the simulations of the nonequilibrium external-force driven flows, such as the gravity, the electric force, and the Lorentz force fields covering all flow regimes.     

FULLY DISCRETE HIGH-ORDER SHOCK-CAPTURING NUMERICAL SCHEMES

The present paper is a sequel to a previous one by the same authors in which a family of up to fourth-order fully discrete (FD) upwind numerical schemes was presented. In this paper we extend those high-order FD schemes to solutions with discontinuities, e.g. shocks. A rigorous anlaysis of the total variation diminishing (TVD) constraint for the high-order FD schemes is carried out. For non-linear systems the TVD constraint is, as usual, applied empirically. These schemes are validated by solving a test problem for the time-dependent shallow water equations.     

用物理黏性构建高阶不振荡对流扩散差分格式

利用数值摄动算法, 通过扩散格式数值摄动重构把对流扩散方程的2阶中心差分格式(2-CDS)重构为高精度高分辨率格式, 解析分析和模型方程计算证实了新格式的高精度不振荡性质. 新格式是把物理黏性使流动光滑化的扩散运动规律引入2-CDS 中的结果. 该法显然与构建高级离散格式的常见方法不同. 证实: 数值摄动重构中引入扩散运动规律的结果格式与引入对流运动规律(下游不影响上游的规律)的结果格式一致, 说明对离散方程的数值摄动运算, 在维持原格式结构形式不动的条件下, 不仅能提高格式精度和稳健性, 且可揭示对流离散运动规律与扩散离散运动规律之间的内在关联;同时证实, 文中提出和使用的上、下游分裂方法是构建高精度不振荡离散格式的一个有效方法.     

基于Boltzmann模型方程的气体运动论统一算法研究

模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程; 研究发展气体运动论离散速度坐标法, 借助非定常时间分裂数值计算方法和NND差分格式, 结合DSMC方法关于分子运动与碰撞去耦技术, 发展直接求解速度分布函数的气体运动论耦合迭代数值格式; 研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法, 由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法. 通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明, 计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好, 证实气体运动论统一算法求解各流域气体流动问题的可行性. 尝试将统一算法进行HPF并行化程序设计, 基于对球体绕流及类``神舟'返回舱外形绕流问题进行HPF初步并行试算, 显示出统一算法具有很好的并行可扩展性, 可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向. 通过将气体运动论统一算法推广应用于微槽道流动计算研究, 已初步发展起可靠模拟二维短微槽道流动数值算法; 通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟, 证实该算法对微槽道气体流动问题具有较强的模拟能力, 可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.      

HIGHER-ORDER SCHEMES FOR FREE SURFACE FLOWS WITH ARBITRARY CONFIGURATIONS

The purpose of the present work was to evaluate the importance of formal accuracy and of the conservation property in the numerical computation of incompressible flows with arbitrary free boundaries, such as occur in wave-breaking problems. Four spatial discretization methods were implemented in a computer code based on the VOF method for tracking free surfaces: a non-conservative four-point scheme, the conservative quadratic upstream interpolation method, the conservative linear extrapolation method and a lower-order conservative scheme based on the power-law discretization. The performance of the four schemes was evaluated in three test problems: the propagation of a solitary wave of high amplitude, the propagation of an undular hydraulic jump and the flow resulting from a breaking hydraulic jump. The main conclusion obtained in the present work was that discrete momentum conservation is more important than the formal accuracy of the spatial discretization scheme, particularly when there is recirculation and breaking.