一类自适应运动网格的ALE方法

从积分形式的二维Lagrange流体力学方程组出发,使用ENO高阶插值多项式,推广了四边形结构网格下的一阶有限体积格式,构造一类结构网格下的高精度有限体积格式.结合有效的守恒重映方法,发展一类高精度的ALE方法,并结合自适应运动网格技术,进行ALE方法的数值模拟,得到预期的效果.     

守恒双曲方程一类本质非振荡(ENO)格式

用自适应Newton插值,结合自适应模型和重构思想去构造数值流通量,对时间采用Runge-Kutta型离散,得到一类不需"真正"插值和数值微分过程的ENO格式。该格式易于数值实现,数值试验表明,这类格式具有良好的计算结果。     

一种混合优化差分格式

优化差分格式一般用于计算气动声学和小尺度的湍流数值模拟,这类格式为了获取更好的短波分辨率通常牺牲了部分收敛精度.文章尝试结合最高阶精度格式与优化格式的特点,构造混合优化格式,提高优化格式的收敛精度以及谱分辨率.混合优化格式由模板上的最高阶精度格式与优化格式加权组合得到,权系数由当前模板上的值确定,这使得该格式为非线性格式.对于单色波问题,通过优化权的设计可大幅度减小相位误差.但是加权混合过程使得计算时间有所增加.数值计算证明了该格式的特点.      

对称TVD差分格式的构造及其应用

本文从ＧＣＩＲ差分格式出发，通过附加中心加权形式的反扩散流量，构造出一类具有二阶精度的对称型ＴＶＤ格式以及相应的一般形式的隐式格式，并针对双曲型方程组问题进行了讨论。同时文中还采用这类格式求解Ｅｕｌｅｒ方程，成功地数值计算了具有激波的流动问题。     

非结构网格下非定常流场双时间步长的加权ENO-强紧致杂交高分辨率格式

针对三维非定常、可压缩流场的Navier-Stokes方程组,本文提出一种新的双时间步长高精度快速迭代格式。该格式在时间上具有二阶精度,在空间离散上不低于三阶。在对流项与粘性项的处理上,本格式分别采用了加权ENO-强紧致格式与紧致四阶精度格式的思想。几个典型算例的实践表明:计算结果与相关实验数据比较吻合,初步表明了该算法可以在非结构网格下具有高效率与高分辨率的特征。     

反应流体的移动网格动理学格式

在移动网格上构造一种反应流的动理学格式.首先利用BGK模型推导含化学反应的流体力学方程组,并利用其积分形式构造移动网格上离散格式,再利用自适应移动网格方法得到网格速度,最后利用时间精确的动理学数值方法构造数值通量,得到移动网格单元上新的物理量.一维与二维的数值实验表明这种格式同时具有高精度、高分辨率的特点.     

基于气动声学问题的高阶非线性紧致格式

文章基于线性中心紧致差分格式, 通过非线性加权插值的方法来求解网格中心处的函数值.这类格式保持了原有中心紧致差分格式的高阶精度和低耗散特性, 同时其分辨率也非常高, 由于其非线性插值的机制, 使得这类格式能够捕捉强激波, 所以这类新的高阶非线性紧致格式是一种较好的模拟湍流和气动声学等多尺度问题的方法.      

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

ENO守恒插值(重映)方法及其在流体计算中的应用

在ENO(Essentially Non-oscillatory)守恒插值方法的基础上,分析和研究现今流体力学计算中涉及的几类网格技术:重叠网格技术、自适应加密技术和运动网格技术.基于ENO插值多项式构造的重映方法具有良好的守恒性,可以有效保证数据传递中物理量的总体守恒.提出该类守恒插值方法在以上几种网格技术中的一些应用前景,并给出一些数值算例.     

二维交错网格的GAUSS型格式

利用Gauss型求积公式在交错网格的情况下构造了一类不需解Riemann问题的求解二维双曲守恒律的二阶显式Gauss型差分格式,该格式在CFL条件限制下为MmB格式.并将格式推广到二维方程组,进行了数值试验.     

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.     

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.     

A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis

A high-order accurate hybrid central-WENO scheme is proposed. The fifth order WENO scheme [G.S. Jiang, C.W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996) 202–228] is divided into two parts, a central flux part and a numerical dissipation part, and is coupled with a central flux scheme. Two sub-schemes, the WENO scheme and the central flux scheme, are hybridized by means of a weighting function that indicates the local smoothness of the flowfields. The derived hybrid central-WENO scheme is written as a combination of the central flux scheme and the numerical dissipation of the fifth order WENO scheme, which is controlled adaptively by a weighting function. The structure of the proposed hybrid central-WENO scheme is similar to that of the YSD-type filter scheme [H.C. Yee, N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys. 150 (1999) 199–238]. Therefore, the proposed hybrid scheme has also certain merits that the YSD-type filter scheme has. The accuracy and efficiency of the developed hybrid central-WENO scheme are investigated through numerical experiments on inviscid and viscous problems. Numerical results show that the proposed hybrid central-WENO scheme can resolve flow features extremely well.     

一种高精度的修正Hermite-ENO格式

设计一种基于三单元具有六阶精度的修正Hermite-ENO格式（CHENO）,求解一维双曲守恒律问题.CHENO格式利用有限体积法进行空间离散,在空间层上,使用ENO格式中的Newton差商法自适应选择模板.在重构半节点处的函数值及其一阶导数值时,利用Taylor展开给出修正Hermite插值使其提高到六阶精度,并设计了间断识别法与相应的处理方法以抑制间断处的虚假振荡;在时间层上采用三阶TVD Runge-Kutta法进行函数值及一阶导数值的推进.其主要优点是在达到高阶精度的同时具有紧致性.数值实验表明对一维双曲守恒律问题的求解达到了理论分析结果,是有效可行的.     

A High-Order WENO Finite Difference Scheme for the Equations of Ideal Magnetohydrodynamics

We present a high-order accurate weighted essentially non-oscillatory (WENO) finite difference scheme for solving the equations of ideal magnetohydrodynamics (MHD). This scheme is a direct extension of a WENO scheme, which has been successfully applied to hydrodynamic problems. The WENO scheme follows the same idea of an essentially non-oscillatory (ENO) scheme with an advantage of achieving higher-order accuracy with fewer computations. Both ENO and WENO can be easily applied to two and three spatial dimensions by evaluating the fluxes dimension-by-dimension. Details of the WENO scheme as well as the construction of a suitable eigen-system, which can properly decompose various families of MHD waves and handle the degenerate situations, are presented. Numerical results are shown to perform well for the one-dimensional Brio–Wu Riemann problems, the two-dimensional Kelvin–Helmholtz instability problems, and the two-dimensional Orszag–Tang MHD vortex system. They also demonstrate the importance of maintaining the divergence free condition for the magnetic field in achieving numerical stability. The tests also show the advantages of using the higher-order scheme. The new 5th-order WENO MHD code can attain an accuracy comparable with that of the second-order schemes with many fewer grid points.     

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes [5], [25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.     

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.     

Local DG method using WENO type limiters for convection–diffusion problems

The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures.