间断有限元方法求解一维非平衡辐射扩散方程

研究一维非平衡辐射扩散方程的数值方法.通过求解间断系数热传导方程的广义黎曼问题,得到一种带加权数值流量,基于该数值流量构造了一类新型的间断有限元方法.在时间离散上采用向后Euler方法,形成的非线性方程组采用Picard迭代求解.数值试验表明该方法具有捕捉大梯度的能力,而且能适应扩散系数间断的情形.     

基于广义交替数值通量的局部间断Galerkin方法求解二维波动方程

波的传播往往在复杂的地质结构中进行,如何有效地求解非均匀介质中的波动方程一直是研究的热点.本文将局部间断Galekin(local discontinuous Galerkin, LDG)方法引入到数值求解波动方程中.首先引入辅助变量,将二阶波动方程写成一阶偏微分方程组,然后对相应的线性化波动方程和伴随方程构造间断Galerkin格式;为了保证离散格式满足能量守恒,在单元边界上选取广义交替数值通量,理论证明该方法满足能量守恒性.在时间离散上,采用指数积分因子方法,为了提高计算效率,应用Krylov子空间方法近似指数矩阵与向量的乘积.数值实验中给出了带有精确解的算例,验证了LDG方法的数值精度和能量守恒性;此外,也考虑了非均匀介质和复杂计算区域的计算,结果表明LDG方法适合模拟具有复杂结构和多尺度结构介质中的传播.     

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.     

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

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.     

一种求解二维辐射流体力学方程组的显隐式格式

构造一种求解二维辐射流体力学方程组的有限体积方法。相较于Euler方程组,辐射流体力学方程组的数值格式设计更为困难。不仅辐射压力与辐射能量的强非线性增加了数值计算的难度,而且求解强激波问题也是一大难点。与此同时,物质量以声速传播,辐射量以光速传播也增加了该系统求解的难度。为了克服这些难点,我们使用MUSCL-Hancock方法求解模型的双曲部分,利用TR/BDF2的变换格式求解其扩散部分。给出了灰非平衡扩散模型关于时间的渐近分析。最后,通过数值实验展现该格式的表现。     

非线性Schrdinger方程的直接间断Galerkin方法

讨论一维和二维非线性Schrdinger(NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schrdinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

一种求解二维辐射流体力学方程组的显隐式格式（英文）

构造一种求解二维辐射流体力学方程组的有限体积方法。相较于Euler方程组,辐射流体力学方程组的数值格式设计更为困难。不仅辐射压力与辐射能量的强非线性增加了数值计算的难度,而且求解强激波问题也是一大难点。与此同时,物质量以声速传播,辐射量以光速传播也增加了该系统求解的难度。为了克服这些难点,我们使用MUSCL-Hancock方法求解模型的双曲部分,利用TR/BDF2的变换格式求解其扩散部分。给出了灰非平衡扩散模型关于时间的渐近分析。最后,通过数值实验展现该格式的表现。     

A Direct Discontinuous Galerkin Method for Nonlinear Schr(o)dinger Equation

讨论一维和二维非线性Schr(o)dinger (NLS)方程的数值求解.基于扩散广义黎曼问题的数值流量,构造一种直接间断Galerkin方法(DDG)求解非线性Schr(o)dinger方程.证明该方法L2稳定性,并说明DDG格式是一种守恒的数值格式.对一维NLS方程的计算表明,DDG格式能够模拟各种孤立子形态,而且可以保持长时间的高精度.二维NLS方程的数值结果显示该方法的高精度和捕捉大梯度的能力.     

非规则形状介质内辐射-导热耦合传热的间断有限元求解

采用间断有限元法(discontinuous finite element method,DFEM)求解非规则形状介质内的辐射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热问题的高精度数值结果.和传统连续型有限元方法不同,DFEM将计算区域划分成相互独立的离散单元,形函数的构造、未知量的加权近似以及控制方程的求解均在每一个离散单元上进行.通过在单元之间施加迎风格式的数值通量,DFEM保证了整个计算区域的连续性,因此这种方法兼具良好的几何灵活性和局部守恒性.推导了辐射传输方程和能量扩散方程的射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热的高精度数值结果.     

预处理JFNK方法求解非平衡辐射扩散方程组

分别采用四种半隐式离散方法构造预处理.针对一维辐射扩散方程组,采用预处理的Jacobian-free Newton-Krylov(PJFNK)求解.数值结果表明预处理方法能够很好地改进JFNK方法的收敛行为.     

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.     

扩散方程的有限方向差分方法

研究二维散乱点集上数值求解非线性扩散方程的有限方向差分方法。利用五个邻点信息构造具有最小模板的离散格式,并且离散系数具有显式表达式。另外,利用五点公式获得了间断问题物质界面的离散格式,该格式对界面流的计算具有近似二阶精度。不同计算区域及不同类型的离散点集上的计算结果验证了方法的有效性。     

Direct discontinuous Galerkin method for the generalized Burgers—Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge-Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.     

Solving coupled nonlinear Schrodinger equations via a direct discontinuous Galerkin method

In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

Solving coupled nonlinear Schrödinger equations via a direct discontinuous Galerkin method

In this work, we present the direct discontinuous Galerkin (DDG) method for the one-dimensional coupled nonlinear Schrödinger (CNLS) equation. We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system. The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method. Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

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.     

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer’s vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer’s. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer’s method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.     

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.