Helbing流体力学交通流模型的守恒形式

得到了Helbing交通流流体力学模型的标准守恒形式,并证明了模型的双曲性,这对研究模型的解析性质和数值格式至关重要.基于给出的守恒形式,设计了高效求解模型方程的LDG(lo-cal discontinuous Galerkin)格式,并模拟了由不稳定平衡态到稳定的时停时走波的演化.数值模拟也表明,通过扩散系数校正确实使模型得到改进,避免了车辆碰撞和出现极端高密度.     

A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.     

DISSIPATIVE NUMERICAL METHODS FOR THE HUNTER-SAXTON EQUATION

In this paper, we present further development of the local discontinuous Galerkin （LDG） method designed in [21] and a new dissipative discontinuous Galerkin （DG） method for the HuntermSaxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].     

非结构网格上一类满足局部极值原理的三阶精度有限体积方法

对二维标量双曲型守恒律方程,发展了一类满足局部极值原理的非结构网格有限体积格式.其构造思想是,以单调数值通量为基础,通过应用基于最小二乘法的二次重构和极值限制器,使数值解满足局部极值原理.为保证数值解在光滑区域达到三阶精度,该格式可结合局部光滑探测器使用.本文从理论上分析了格式的稳定性条件,数值实验验证了格式的精度和对间断的分辨能力.     

多孔介质中单相气体局部流动的均质化建模

该文研究了渐近均质法在单相气体渗流理论中的应用,开发了气体在孔隙尺度下流动的数学模型和数值方法.基于渐近均质法,建立了周期单元上描述周期性多孔结构孔隙尺度下单相气体流动的局部问题.讨论了局部问题的特殊数学性质和物理意义.利用一种基于对称性和反对称性扩展的简化方法,提出了求解局部问题的最小二乘有限元方法,克服了由于平均算子和周期性边界条件引起的数值困难.局部问题的求解能够获得单孔内速度和压力的精确分布,并且在仅知道孔隙几何形状的情况下评估多孔介质的渗透性.在局部问题的基础上,通过理论分析获得了微管中Poiseuille流动的解析解,验证了所提出的数学模型和数值算法.最后,考虑了一种三维周期性多孔结构,获得了单孔中气体局部流动的数值结果和多孔介质的渗透系数.     

A conservative local discontinuous Galerkin method for the solution of nonlinear Schrdinger equation in two dimensions

In this study, we present a conservative local discontinuous Galerkin(LDG) method for numerically solving the two-dimensional nonlinear Schrdinger(NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor(IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.     

不可压饱和多孔弹性杆动力响应的多辛方法

研究了不可压饱和多孔弹性杆的一维动力响应问题.基于多孔介质理论,在流相和固相微观不可压、固相骨架小变形的假定下,建立了不可压流体饱和多孔弹性杆一维轴向动力响应的数学模型.利用Hamilton空间体系的多辛理论,构造了不可压饱和多孔弹性杆轴向振动方程的多辛形式及其多种局部守恒律.采用中点Box离散方法得到轴向振动方程的多辛离散格式和局部能量守恒律以及局部动量守恒律的离散格式;数值模拟了不可压饱和多孔弹性杆的轴向振动过程,记录了每一时间步的局部能量数值误差和局部动量数值误差.结果表明,已构造的多辛离散格式具有很高的精确性和较长时间的数值稳定性,这为解决饱和多孔介质的动力响应问题提供了新的途径.     

带Markov跳时变随机种群收获系统数值解的收敛性

讨论了一类带Markov跳时变随机种群收获系统的数值解问题.利用EulerMaruyama方法给出了时变种群系统的数值解表达式,在局部Lipschitz条件下,证明了方程的数值解在均方意义下收敛于其解析解.最后,通过数值例子对所给出的结论进行了验证.     

三维尾迹型流动中的大尺度旋涡位错形成

应用紧致有限差分——Fourier谱方法求解三维不可压Navier-Stokes方程组,数值研究尾迹型流动的三维演化和旋涡位错. 计算中在来流剖面中引入局部展向非均匀性.数值结果表明流动的不稳定性发展导致三维旋涡涡街流场. 局部非均匀性引起涡街间（沿展向）在频率、相位及强度上的差异. 在非均匀区中产生并形成大尺度链状旋涡位错结构. 用数值模拟详细描述了旋涡位错的形成与特征.     

求解Hamilton-Jacobi方程的局部移动网格方法

对Hamilton-Jacobi方程设计了一个基于Runge-Kutta间断Galerkin方法的移动网格方法,并利用坏单元指示子进一步设计了一个局部移动网格方法.数值结果表明这两个方法相比均匀网格能提高数值解的质量.同时局部移动网格方法通过将网格移动局部化,在不影响精度的前提下节省了计算时间,提高了计算效率。