摄动有限体积法重构近似高精度的意义

研讨有限体积(FV)方法重构近似高精度的作用问题.FV方法中积分近似采用中点规则为二阶精度时,重构近似高精度(精度高于二阶)的意义和作用是一个有争议的问题.利用数值摄动技术[1,2]构造了标量输运方程的积分近似为二阶精度、重构近似为任意阶精度的迎风型和中心型摄动有限体积(PFV)格式.迎风PFV格式无条件满足对流有界准则(CBC),中心型PFV格式为正型格式,两者均不会产生数值振荡解.利用PFV格式求解模型方程的数值结果表明:与一阶迎风和二阶中心格式相比,PFV格式精度高、对解的间断分辨率高、稳定性好、雷诺数的适用范围大,数值地"证实"重构近似高精度和PFV格式的实际意义和好处.     

间断伽辽金方法在燃机空气系统中的应用

间断伽辽金方法(DGM)对守恒变量及梯度变量具有一致的任意高阶计算精度,因此该方法也具有较高的热流计算精度。本文搭建了二次空气系统单元机理实验台,并采用DGM对中心进气转静系开展了数值计算,取得了误差小于10%的对流换热系数计算结果。此外本文结合SST-γ-Re_θ转捩模型,应用有限体积方法(FVM)及DGM,对燃气轮机旋转盘腔内流动与传热进行了预测,并取得了与实验结果吻合较好的传热预测结果。此两转转系及转静系典型问题验证了DGM对燃气轮机二次空气冷却流动传热预测具有一定优势,展现了DGM在燃气轮机二次空气系统中广阔的应用前景。     

一维守恒的Lagrangian ADER格式

基于欧拉框架下ADER格式,构造一维守恒只有一个时间步的、高精度中心型拉格朗日ADER(LADER)格式.构造r阶LADER格式包括:从欧拉方程出发推导拉格朗日框架下积分形式的方程、采用WENO方法高精度重构节点处守恒量和从1阶到r-1阶的空间导数、求拉氏框架下这些变量的Godunov值,并计算1阶到r-1阶的时间全导数,最后高精度离散积分形式的流通量函数.对光滑流场的模拟表明,LADER格式达到设计的精度;对含强间断的流场模拟表明,数值解在间断附近基本无振荡.     

间断Galerkin方法求解跨音压气机转子流场

本文采用间断Galerkin方法求解雷诺平均N-S方程和SA湍流模型方程,采用Roe格式计算无黏通量,混合格式求解黏性通量中的梯度。为防止跨音速流场计算中的数值振荡,采用了TVD限制器和正性修正。计算了跨音速转子NASA Rotor37在设计转速下的变工况流场,得到了与实验吻合良好的结果,表明DG方法在叶轮机械内部流动计算中具有广阔的应用前景。     

混合稀疏水声信道的动态区分性压缩感知估计

由于水声传播过程中同时存在声信号直达、静态或动态边界反射的现象,水声信道会呈现不同动态特性的多径,形成具有混合稀疏的结构,即多径由静态或相对缓变的平稳多径分量和快速时变的动态多径分量混合组成。对于混合稀疏信道,经典的稀疏信道估计算法未考虑混合稀疏性,将导致算法失配、性能下降;以时变稀疏集为模型,动态压缩感知(DCS)结合卡尔曼滤波(KF-CS)可提高对时变多径分量的估计精度,但KF对静态稀疏分量的估计无法充分挖掘其稀疏性。通过将混合稀疏水声信道建模为由静态和时变支撑集所组成的稀疏集,提出一种动态区分性压缩感知(DDCS)方法。该算法首先结合同步正交匹配追踪(SOMP)和正交匹配追踪(OMP)将混合稀疏多径进行区分,分解为静态分量和时变分量;然后,分别用KF-CS和同步正交匹配追踪算法估计时变和静态多径的幅度;最后,将静态分量和时变分量的估计结果整合以得到整个水声信道的冲激响应。通过海试实验把所提DDCS算法与经典信道估计算法、压缩感知算法和DCS算法进行了比较,验证了所提算法的有效性。结果表明,对混合稀疏水声信道进行区分性稀疏估计可改善信道估计性能,进而可通过信道估计均衡器提升水声通信质量。      

一种基于高阶节点间断Galerkin方法的新型限制器设计

使用高阶间断Galerkin(discontinuous Galerkin, DG)方法求解双曲守恒律方程组时, 非物理效应常常导致计算过程的中断, 这在很大程度上制约着该方法在计算流体力学中的应用.文章结合局部单元上原始流动变量的Taylor展开, 设计了一种新型的限制器, 通过对各阶空间导数的重构, 有效地消除了非物理振荡的不利影响.对二维Euler方程的计算结果表明, 该限制器不仅能够捕捉高质量的激波, 而且能够保证残值的有效收敛.      

格子Boltzmann方法分布函数重构算子的应用

给出了格子Boltzmann方法(LBM)密度分布函数和温度分布函数的两个重构算子,解决了LBM与其它方法耦合的关键难题.构造了LBM与有限容积法(FVM)以及LBM与分子动力模拟方法(MD)的两个耦合模型.通过方腔自然对流以及平板通道内的泊肃叶流动对耦合模型进行验证.结果表明,基于重构算子的耦合模型可以正确地应用于L...     

三维空间二阶精度流体体积界面重构方法

给出三维空间网格模板含81个单元的最小二乘流体体积界面重构方法,并和Youngs方法及网格模板含125个单元的最小二乘流体体积界面重构方法进行比较.静态和动态的测试例子均表明:该方法能精确重构任意方向的平面界面,对C²光滑曲面它能达到二阶收敛精度.和网格模板含125个单元的最小二乘流体体积界面重构方法相比,在达到同样网格精度的条件下,减少了计算量,节省了计算时间,提高了计算效率.     

二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

针对二维非定常半线性扩散反应方程，空间导数项采用四阶紧致差分公式离散，时间导数项采用四阶向后Euler公式进行离散，提出一种无条件稳定的高精度五层全隐格式.格式截断误差为O（τ⁴+τ²h²+h⁴），即时间和空间均具有四阶精度.对于第一、二、三时间层采用Crank-Nicolson方法进行离散，并采用Richardson外推公式将启动层时间精度外推到四阶.建立适用于该格式的多重网格方法，加快在每个时间层上迭代求解代数方程组的收敛速度，提高计算效率.最后通过数值实验验证格式的精确性和稳定性以及多重网格方法的高效性.     

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。     

A class of hybrid DG/FV methods for conservation laws I: Basic formulation and one-dimensional systems

By comparing the discontinuous Galerkin (DG) and the finite volume (FV) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ is introduced for high-order numerical methods. Based on the new concept, a class of hybrid DG/FV schemes is presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of a piecewise polynomial solution are computed locally in a cell by the DG method based on Taylor basis functions (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and its adjacent neighboring cells. The hybrid DG/FV methods can greatly reduce CPU time and memory required by the traditional DG methods with the same order of accuracy on the same mesh, and they can be extended directly to unstructured and hybrid grids in two and three dimensions similar to the DG and/or FV methods. The hybrid DG/FV methods are applied to one-dimensional conservation law, including linear and non-linear scalar equation and Euler equations. In order to capture the strong shock waves without spurious oscillations, a simple shock detection approach is developed to mark ‘trouble cells’, and a moment limiter is adopted for higher-order schemes. The numerical results demonstrate the accuracy, and the super-convergence property is shown for the third-order hybrid DG/FV schemes. In addition, by analyzing the eigenvalues of the semi-discretized system in one dimension, we discuss the spectral properties of the hybrid DG/FV schemes to explain the super-convergence phenomenon.     

A class of hybrid DG/FV methods for conservation laws II: Two-dimensional cases

By comparing the discontinuous Galerkin (DG) methods, the k-exact finite volume (FV) methods and the lift collocation penalty (LCP) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ was introduced for higher-order numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods was presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent face neighboring cells. In this follow-up paper, the hybrid DG/FV schemes are extended onto two-dimensional unstructured and hybrid grids. The two-dimensional linear and non-linear scalar conservation law and Euler equations are considered. Some typical cases are tested to demonstrate the performance of the hybrid DG/FV method, and the numerical results show that they can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy in the same mesh.     

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.     

Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms

In [16], [17], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions), it is more difficult to design high order schemes which do not produce negative density or pressure. In this paper, we first show that our framework to construct positivity-preserving high order schemes in [16], [17] can also be applied to Euler equations with a general equation of state. Then we discuss an extension to Euler equations with source terms. Numerical tests of the third order Runge–Kutta DG (RKDG) method for Euler equations with different types of source terms are reported.     

On maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.     

Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., [6], [10]. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in [1], [2] can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in [22]. In this paper, we first discuss an extension of the technique in [22], [23], [24] to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in [22]. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.     

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system

Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community , , , , , ,  and . In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [6], as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge–Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter [37], which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method [23]. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.     

Positivity-preserving high order finite difference WENO schemes for compressible Euler equations

In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.     

轴对称钝体后湍流扩散燃烧的PDF模拟

在无结构网格中，对轴对称钝体驻定的湍流扩散火焰进行数值模拟．采用有限容积(FV)／Monte Carlo(MC)混合算法求解湍流燃烧问题的混合算法．Monte Carlo法求解脉动速度一标量．频率的联合概率密度函数方程，有限容积法求解平均质量、动量和能量方程．求解的两组方程是相容的，合理的耦合方式可以减少统计偏差，计算精度和效率显著优于单独的颗粒方法．文中对化学反应采用层流火焰面模型，并将数值计算结果与实验结果作了比较和分析．     

二维浅水波方程的数值激波不稳定性

研究二维浅水波方程的数值激波不稳定性问题.线性稳定性分析和数值实验表明,格式的临界稳定性与数值激波的不稳定现象有重要的联系.基于扰动量的增长矩阵分析,本文将高分辨率的数值格式和HLL格式进行特定的加权,设计一类新的混合型数值格式.其中可以调节非线性波速的HLLC与HLL的混合格式,数值试验展示了消除浅水波方程激波不稳定现象的有效性和鲁棒性.