任意马赫数非定常流动数值模拟的统一算法

发展适用于从低速到高速任意马赫数非定常流动数值模拟的统一算法.通过引入一个伪时间导数项和一个新的预处理矩阵,得到双时间非定常预处理可压缩Navier-Stokes方程.方程的对流项采用三阶Roe通量近似差分格式离散,粘性项采用二阶中心差分格式离散.基于数值通量的线性化技术,实现伪时间步的隐式ADI-LU格式迭代,进而获得物理时间步的二阶推进精度.重点以低马赫数流动为例,求解了圆柱绕流和NACA0015翼型等速上仰动态失速问题.计算结果表明该统一算法能够较好地模拟低马赫数乃至任意马赫数非定常流动.     

隐式格式求解拟压缩性非定常不可压Navier-Stokes方程

采用Rogers发展的双时间步拟压缩方法,数值求解不可压非定常问题.数值通量分别采用三阶精度Roe格式和二阶精度Harten-Yee的TVD格式离散.为了加快收敛,提高求解效率,试验了几种隐式格式(ADI-LU,LGS,LU-SGS).针对经典的低雷诺数(Re=200)圆柱绕流问题,比较了不同隐式方法的计算结果和求解效率,以及两种数值离散格式计算结果的异同.最后采用Roe格式数值求解了两种典型的低速非定常流动问题:绕转动圆柱(ω=1)低雷诺数流动;NACA0015翼型等速拉起数值模拟.     

含双时间步法的化学非平衡流解耦算法

发展基于隐式双时间步法的化学非平衡流解耦型计算方法.采用算子分裂法对流动和反应进行解耦处理,流动方程组通过双时间步方法求解;源项方程组采用二阶梯形公式迭代求解;提出"源项消去"法,以消除化学反应源项对流动求解引入的误差,从而保证流动方程组求解的时间精度.理论分析和计算结果表明,方法既可以保证双时间步法的求解效率,又可以获得比较精确的非定常计算结果.     

非定常流动的隐式间断Galerkin方法研究

为了增加间断Galerkin(Discontinuous Galerkin,DG)方法在非定常流动中的求解效率,本文开展了非定常流动的隐式DG方法研究。隐式DG方法的构造采用二阶向后差分格式(BDF2)进行时间项离散,非线性代数系统的求解基于Newton迭代法,采用块对称Gauss-Seidel(SGS)迭代法对线性方程组进行了求解。基于所发展的非定常流动的隐式DG方法,分别对等熵圆柱扰流和卡门涡街(Re=100)现象进行了数值模拟。研究结果表明,所发展的隐式DG方法能够达到设计精度,能够在高出显式方法两个数量级的时间步长上保持稳定,具有高的求解效率,且计算结果与显式方法和相关文献均吻合较好。     

基于界面修正的迭代并行计算方法

构造基于界面修正的迭代并行方法的一般途径是：将物理空间区域剖分成若干不重叠的块；在分块子区域的内边界上，采用某种显式格式计算出界面值作为预估值；然后采用某种隐式格式并行求解各个子块区域上的解，这里的隐式格式通常需要进行迭代求解(称为内迭代)；可在每一迭代步或几次迭代步结束时，利用已计算出的分块子区域内的(近似)解，在分块子区域内边界处利用隐式格式计算出在内边界处的校正值；随后再转入各个子块区域上的求解，该过程称为外迭代。与以往的并行差分格式不同，在求解的子区域上的定解问题时，可以仅仅在第一个(初始)迭代步求解时所需边界条件使用子区域内界面处的某种显式格式的解，在随后的迭代步中即可改用子区域内界面处的隐式修正格式的解。由此，至少可区分如下3类性质不完全相同的迭代并行格式。     

谱方法用于非定常流动计算的隐式求解

本文对谱方法用于周期性非定常流动的隐式求解方法进行了探讨,分析了影响计算稳定性和收敛速度的因素.提出了结合多重网格的隐式求解方法并对算法进行了验证,初步计算表明本文算法具有良好的稳定性和收敛速度.对于周期性非定常流动,结合本文提出的隐式求解的时域谱方法可以达到很高的精度且具有良好的计算效率.     

一种隐式预处理方法及其在定常和非定常流动数值模拟中的应用

将Choi-Merkle矩阵预处理方法与LU-SGS隐式方法、双时间法以及多重网格方法结合,发展适用于绕飞行器定常和非定常粘性流动的高效隐式预处理计算方法和程序.介绍一种针对定常和非定常流动的LU-SGS隐式预处理方法的统一表述方法.在不改变流动解的前提下,对Navier-Stokes方程的伪时间导数项实施Choi-Merkle矩阵预处理,从而改善可压缩控制方程在低速情况下的系统刚性,使基于LU-SGS时间推进格式的数值模拟方法同时适用于从极低马赫数到可压缩范围内的数值模拟.对Jameson中心格式的人工粘性进行相应的修改,以提高低速流动的计算精度.翼型、机翼以及翼身组合体绕流的数值模拟研究表明,隐式预处理方法获得了很高的计算效率,可使马赫数0.1左右的低速流动计算时间减少50%以上;通过对现有可压缩计算程序进行小量改动,便可使其均匀覆盖整个低速流动范围,提高CFD程序在飞行器绕流数值模拟中的实用性.     

2阶精度时间后差隐式格式TVD充分条件

非定常流动问题计算中常用到含三个时间层的二阶精度时间后差隐式格式,并且希望构成TVD格式,然而理论上的问题多年一直没有解决。本文找到了解决办法,构造了这种类型的隐式TVD格式,证明了其为TVD的充分条件。理论结果为计算所验证,并表明通常未采取本文对时间差分处理方法的格式尚不具备TVD性质。     

非定常对流扩散方程的高精度多重网格方法

由已有的求解定常对流扩散方程的高阶紧致差分格式出发,直接推导出了数值求解非定常对流扩散方程的一种高阶隐式紧致差分格式,其时间为二阶精度,空间为四阶精度,并且是无条件稳定的。为了加快传统迭代法在求解隐格式时在每一个时间步上的迭代收敛速度,采用了多重网格加速技术。数值实验结果验证了本文方法的高阶精度、高效性及高稳定性。     

应用隐式SMAC格式计算射流放水阀内部非定常不可压粘性流场

为了分析射流放水阀内部的非定常不可压粘性流场,在一般曲线坐标系下求解了以逆变速度为参数的不可压Ｎ－Ｓ方程和压力椭圆方程。计算采用了基于隐式ＳＭＡＣ格式的有限差分格式。在这种格式中,非定常流场的速度是通过使用Ｃｒａｎｋ－Ｎｉｃｈｏｌｓｏｎ格式,并在每一时间步处以牛顿迭代的方法得到的;而压力椭圆方程是通过使用Ｔｓｃｈｅｂｙｓｃｈｅｆｆ ＳＬＯＲ格式并交替改变计算方向而求解。通过使用交错网格和迎风差分,诸如改进的ＱＵＩＣＫ格式;计算过程中的误差和数值不稳定能够得到有效地控制。计算结果和实验观察得到的流动图谱对比表明,二者十分吻合。     

MCore: A non-hydrostatic atmospheric dynamical core utilizing high-order finite-volume methods

This paper presents a new atmospheric dynamical core which uses a high-order upwind finite-volume scheme of Godunov type for discretizing the non-hydrostatic equations of motion on the sphere under the shallow-atmosphere approximation. The model is formulated on the cubed-sphere in order to avoid polar singularities. An operator-split Runge–Kutta–Rosenbrock scheme is used to couple the horizontally explicit and vertically implicit discretizations so as to maintain accuracy in time and space and enforce a global CFL condition which is only restricted by the horizontal grid spacing and wave speed. The Rosenbrock approach is linearly implicit and so requires only one matrix solve per column per time step. Using a modified version of the low-speed AUSM⁺-up Riemann solver allows us to construct the vertical Jacobian matrix analytically, and so significantly improve the model efficiency. This model is tested against a series of typical atmospheric flow problems to verify accuracy and consistency. The test results reveal that this approach is stable, accurate and effective at maintaining sharp gradients in the flow.     

Adaptive higher-order finite element methods for transient PDE problems based on embedded higher-order implicit Runge–Kutta methods

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combine adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge–Kutta methods in time. Weak formulation is only created for the stationary residual, and the Runge–Kutta methods are specified via their Butcher’s tables. Around 30 Butcher’s tables for various Runge–Kutta methods with numerically verified orders of local and global truncation errors are provided. A time-dependent benchmark problem with known exact solution that contains a sharp moving front is introduced, and it is used to compare the quality of seven embedded implicit higher-order Runge–Kutta methods. Numerical experiments also include a comparison of adaptive low-order FEM and hp-FEM with dynamically changing meshes. All numerical results presented in this paper were obtained using the open source library Hermes (http://www.hpfem.org/hermes) and they are reproducible in the Networked Computing Laboratory (NCLab) at http://www.nclab.com.     

Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit–explicit Runge–Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier–Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge–Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge–Kutta methods.     

Optimal Runge–Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge–Kutta time integrators, with the aim of deriving optimal Runge–Kutta schemes for wave propagation applications. We review relevant Runge–Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge–Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge–Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.     

A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two-dimensions

This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.     

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.     

A finite volume implicit time integration method for solving the equations of ideal magnetohydrodynamics for the hyperbolic divergence cleaning approach

A finite volume numerical technique is proposed to solve the compressible ideal MHD equations for steady and unsteady problems based on a quasi-Newton implicit time integration strategy. The solenoidal constraint is handled by a hyperbolic divergence cleaning approach allowing its satisfaction up to machine accuracy. The conservation of the magnetic flux is computed in a consistent way using the numerical flux of the finite volume discretization. For the unsteady problem, the time accuracy is obtained by a Newton subiteration at each physical timestep thereby converging the solenoidal constraint to steady state. We perform extensive numerical experiments to validate and demonstrate the capabilities of the proposed numerical technique.     

叶栅内定常及非定常粘性流动数值模拟

本文给出了一个模拟叶栅内准三维定常和非定常粘性流动的数值方法。对于定常流动,采用ＴＶＤ Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式和代数湍流模型求解雷诺平均Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程,使用当地时间步长和多网格技术使计算加速收敛到定常状态;对于非定常流动,使用双时间步长和全隐式离散,采用与求解定常流动相似的多网格方法求解隐式离散方程。文中给出了ＶＫＩ透平叶栅内的定常流结果和１．５级透平叶栅内的非定常数值结果。     

On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations

Finite element and finite difference discretizations for evolutionary convection–diffusion–reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank–Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge–Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.