一维多介质可压缩Euler方程的高精度RKDG有限元方法

采用RKDG有限元目的、Level Set目的和改进的带"Isentropic"修正的Ghost Fluid目的模拟了一维多介质可压缩Euler方程,其中Euler方程、Level Set方程和重新初始化方程都采用了三阶精度的RKDG有限元目的进行离散,并对一维两种介质可压缩流体进行了数值实验,得到了较高分辨率的计算结果.     

结合KFVS方法的RKDG有限元格式求解可压缩流体力学方程组

迎风格式被应用于双曲守恒率方程组求解时，往往需要计算精确或近似的Riemann解，这无疑增加了数值算法的复杂性。为了能降低迎风格式的这一复杂性，直接对宏观通矢量进行分裂的迎风算法—通矢量分裂算法FVS(Flux Vector Splitting)的研究引起了人们的广泛关注。第一个通矢量分裂算法是Sanders和Prendergast基于气体分子动力学理论提出的Beam格式。由于Beam格式构造的基础是用3个Delta函数逼近分子速度平衡分布函数Maxwell分布函数，因而格式的数值耗散仍然比较大。类似于Beam格式，     

用代数多重网格法求解一维分裂格式的Euler方程

提出了代数多重网格法(AMG)的一种新算法。新算法改进了插值公式和粗网格方程,并把它应用到求解一维的分裂格式Euler方程。数值结果表明,对于具有高CFL条件数的Euler方程,代数多重网格法可以求解;对于Gaus-Seidel方法求解不能收敛的代数方程组,代数多重网格法求解可以收敛。新算法改进了代数多重网格法的收敛性和扩展了它的应用范围,数值结果表明了它的有效性和强壮性。     

求解Euler方程的高精度的空间--时间守恒格式

空间-时间守恒(STC)格式是近年来发展出的一种计算格式,在现有的STC格式构造过程中,流动变量在解元中的分布都用其一阶Taylor展开式来表示.STC格式的精度与所采用的Taylor展开式的阶数有关.该文采用流动变量的二阶Taylor展开式来表示其在解元上的分布、构造出了求解一维Euler方程的STC格式.用该格式对几个问题进行了计算,将计算结果与精确解进行了比较,比较表明该格式有较高的精度.     

求解Euler方程的空间—时间守恒格式

本文在CE／SE方法的基础上，提出了空间－时间守恒（STC）格式，其特点是构造简单，物理概念清晰，守恒性好，计算速度快且精度高，容易推广到多维流动及粘性流动。通过对二维Euler方程STC格式的介绍，可以看出这一方法的主要特点。与其它格式的计算结果或精确解相比较表明，用STC格式计算的结果是令人满意的。     

求解Euler方程的区域分解方法与并行算法

将复杂形状区域划分成多块子区域,研究发展了一种多块区域之间迎风守恒型的内边界耦合方法,实现相邻子区域解的光滑过渡,使多区耦合得到总体流场的数值解。对二维翼型跨音速流动和圆弧形隆起物超音速流动等进行了分区数值计算,并将计算结果与单区计算结果和实验结果作了比较。并行分区计算引入"先进先出"的同步控制等待机制,实现了高效率并行计算,还分析了影响并行效率的主要因素。     

利用拉格朗日量求解黑洞背景下自由光子的运动

利用拉格朗日量求解力学问题是一种很有用的方法.本文将借助于拉格朗日量求解黑洞时空背景下自由光子的运动,在求得光子的四速度之后,借助于机械能守恒定律,考察具有不同能量的光子在黑洞背景下所允许的运动范围.随后根据力学中运动学反问题,最终得到光子的运动轨迹.     

求解二维Euler方程有限单元边插值的降维重构算法

数值求解二维Euler方程的有限体积法（如k-exact，WENO重构、紧致重构等），无一例外地要进行耗时的网格单元上的二维重构.然而这些二维重构最后仅用于确定网格单元边界上高斯积分点处的解值，单元上二维重构似乎并非必需的.因此，文章提出用网格边上的一维重构来取代有限体积法中网格单元上的二维重构，分别在一致矩形网格和非结构三角形网格上发展了基于网格边重构的求解二维Euler方程的新方法，称为降维重构算法.数值算例表明该算法可以计算有强激波的无黏流动问题，且有较高的计算效率.      

非结构网格RKDG有限元方法求解多介质界面问题

在流体力学方程数值模拟中，多介质界面的计算是一个非常重要的问题。已有的数值模拟多采用Lagrange方法。但是，当计算长时间问题时，Lagrange方法会产生网格扭曲，进而使计算无法进行下去。另外一种是采用Euler方法，Euler方法应用于多介质流体力学，首要问题是界面捕捉，即将不同区域介质界面描述清楚，这样才能使得多介质流体计算时，不会发生流体的非物理振荡。     

Lagrange坐标系下二维气动方程组的RKDG有限元方法

构造Lagrange坐标系下二维可压缩气动方程组的RKDG(Runge-Kutta Discontinuous Galerkin)有限元方法.将流体力学方程组和几何守恒律统-求解,所有计算都在固定的网格上进行,计算过程中不需要网格节点的速度信息.对几个数值算例进行数值模拟,得到较好的数值模拟结果.     

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

二维多介质可压缩流的RKDG有限元方法

应用RKDG(Runge-Kutta Discontinuous Galerkin)有限元方法、Level Set方法和Ghost Fluid方法数值模拟二维多介质可压缩流,其中Euler方程组、Level Set方程和重新初始化方程的空间离散采用DG(Discontinuous Galerkin)有限元方法,时间离散采用Runge-Kutta方法.对二维的气-气和气-液两相流进行了数值计算,得到了分辨率较高的计算结果.     

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin(RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ?ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ?ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite element method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discontinuous Galerkin (RKDG) method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.     

二维Lagrangian坐标系下可压气动方程组的间断Petrov-Galerkin方法

构造矩形网格下求解Lagrangian坐标系下气动方程组的单元中心型格式. 空间离散采用控制体积间断Petrov-Galerkin方法,时间离散采用二阶TVD Runge-Kutta方法. 利用限制器来抑制非物理震荡并保证RKCV算法的稳定性. 构造的算法可以保证物理量的局部守恒. 与Runge-Kutta间断Galerkin（RKDG）方法相比较,RKCV方法的计算公式少一项积分项使得计算较简单. 给出一些数值算例验证了算法的可靠性及效率.     

A high order ENO conservative Lagrangian type scheme for the compressible Euler equations

We develop a class of Lagrangian type schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially non-oscillatory (ENO) reconstruction. They are conservative for the density, momentum and total energy, can maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One and two-dimensional numerical examples in the Cartesian and cylindrical coordinates are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties.     

保持对称性的二维Lagrange流体力学有限差分方法

广泛应用的二维直角坐标系下的Wilkins有限差分格式在计算一维柱面问题时,通过等角度划分周向网格能够获得严格的对称性,非等角度划分周向网格会产生较严重的不对称性.通过分析Wilkins有限差分格式在处理非等角度划分周向网格的一维柱面问题时破坏对称性的原因,指出周向网格的非等角度划分产生了周向压力分量,从而产生了周向加速度分量和周向运动速度,以此为基础提出一种对该有限差分格式进行修正的方法,将节点处的周向压力分量做算术平均运算,以消除周向压力分量,只剩径向压力分量起作用.因而该修正方法在以任意角度划分周向网格的条件下都能够保持严格的对称性.通过几个典型算例验证该结论,对对称流动,修正方法与原始方法所获得的结果一致,对非对称流动,二者有微小差异.