非定常自由面流激波解的二阶守恒算法

将计算双曲型守恒律弱解的Lax-Wendroff型TVD格式推广到断面形状沿程任意变化的一般浅水方程组，构造了二阶精度的差分格式．新格式适用于模拟天然河道中溃坝洪水波的传播．提供了表明方法性能的算例，实际天然梯级水库溃坝问题的数值实验表明格式稳定，适应性强．     

有限体积KFVS方法在二维溃坝中的应用

本文采用了基于KFVS格式的有限体积方法 (FVM)求解了控制水流运动的二维浅水方程 ,建立了二维水坝瞬间溃坝的洪水演进模型 .并应用此模型模拟了二维非对称溃坝和对称溃坝情形下坝左下角有障碍物时的洪水波演进过程 .模拟结果表明该数学模型对二维浅水运动的模拟很有效 .     

基于故障树分析的堰塞湖溃坝概率估计方法

堰塞湖排险的一个关键问题是如何针对实施不同应对措施情况下的堰塞湖溃坝概率进行估计,这是一个值得关注的重要研究课题。本文提出了一种基于故障树分析(FTA)的堰塞湖溃坝概率估计方法。首先,通过堰塞湖排险问题的实际背景分析,基于FTA构建了堰塞湖溃坝故障树的基本架构;然后,通过相关领域知识、历史案例分析、专家主观判断和多位专家主观判断信息的融合,可以确定实施某一应对措施情形下故障树中各基本事件在不同时段内发生的概率;进一步地,依据构建的故障树和基本事件发生的概率,给出了在不同时段内堰塞湖溃坝事件发生的概率的估计方法。最后,通过一个实例分析说明了本文所提出方法的可行性与有效性。     

唐家山堰塞湖泄洪问题的研究

研究的是唐家山地震次生灾害引发的堰塞湖问题.首先对数字高程地图进行等高图像分析求解了堰塞湖不同高程水位对应的湖区面积,建立了蓄水量体积与堰塞湖水位高程的离散化模型,然后建立了神经网络模型和多元线性回归模型研究了北川降雨量与堰塞湖入库流量的关系,继而求解得到不同降雨量下每日堰塞湖水位高程.在研究泄洪过程时,首先通过对泄洪过程和溃坝过程内在机理的研究分别建立了正交多项式逼近模型和仿真模型得到溃坝时的溃口流量随时间变化的关系,继而分析求解得到溃坝时其他参数随时间变化的关系.针对淹没区的问题,综合数字高程地图和行政区域地图,利用数字地图计算了洪水到达各被淹没区域的时间,淹没范围,以便于确定撤离方案.     

修正压力梯度粒子近似SPH方法计算大密度比界面流动

计算了高密度比的多界面流动问题.为保证多相SPH(smoothed-particle hydrodynamics)方法捕捉界面光滑性和消除界面附近压力震荡，修正了动量方程压强梯度项的粒子近似，在界面施加了排斥力.采用Rayleigh-Taylor界面不稳定性、非Boussinesq锁定交换、溃坝和气泡上升等算例验证了该方法的准确性和健壮性，得到不同时刻界面（粒子）分布、压力云图和指定点压力时间分布、界面锋面距离等.所得结果表明:计算结果(如界面形状、光滑性和指定点压力分布等)与实验值或其他文献结果符合较好.修正的压力梯度项粒子近似，改善了多相SPH方法对高密度比、大变形和破碎多相界面的模拟能力和光滑性，同时界面附近未出现明显的压力震荡.     

唐家山堰塞湖泄洪问题研究

研究了唐家山堰塞湖的泄洪规律,并对其蓄水、溃坝、泛洪等三个过程进行了模型分析.     

一维浅水方程的格子Boltzmann模拟

通过无量纲化变换 ,建立了一个一维浅水方程的LB模型 ,对一维溃坝波进行了模拟计算 ,并分别与文献结果和精确解作了比较 ,说明了此种模型的进步之处和可行性 .     

Banach空间中含间断项Sturm—Liouville问题的解

利用拟上下解方法和混合单调迭代法,研究了Banach空间中含间断项的非线性Sturm-liouville问题解的存在唯一性,并给出逼近解迭代序列的误差估计．     

重构电导率间断界面的一种水平集方法

对一类EIT（电阻抗断层成像）问题提出一种水平集方法来重构电导率的间断界面．通过选取适当速度函数,构造了水平集重构算法,同时给出EIT问题及正则化的理论结果．数值例子表明重构算法是有效和稳定的．     

The Hilbert Problem for the First Order Elliptic Complex Equation With Discontinuous Coefficients

本文研究一阶带间断系数的椭圆型复方程的Ｈｉｌｂｅｒｔ边值问题，在一定条件下给出了上述问题解的存在性、可解条件以及解的积分表示式．     

Spectral problems for variational inequalities with discontinuous operators

Some spectral problems for variational inequalities with discontinuous nonlinear operators are considered. The variational method is used to prove the assumption that such problems are solvable. The general results are applied to the corresponding elliptic variational inequalities with discontinuous nonlinearities.     

求解椭圆边值问题惩罚形式的间断有限元方法

本文提出了一个新的求解二阶椭圆边值问题的惩罚形式间断有限元方法并给出了稳定性和收敛性分析. 特别地,本文建立了间断有限元解的基于余量的后验误差估计,给出了求解间断有限元方程的自适应算法.       

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.     

Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

In this paper, we propose and analyze a fully discrete local discontinuous Galerkin (LDG) finite element method for time-fractional fourth-order problems. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Stability is ensured by a careful choice of interface numerical fluxes. We prove that our scheme is unconditional stable and convergent. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

Nonlinear boundary value problems for discontinuous delayed differential equations

In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.     

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

In this paper, a discontinuous Galerkin least-squares finite element method is developed for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. In a companion paper (Lin in SIAM J Numer Anal 47:89–108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.     

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.     

Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles. This approach leads to optimal convergence in the space and time discretization parameters.     

Existence and approximation of solutions for discontinuous functional differential equations

In this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly. By applying the method of upper and lower solutions, which may be discontinuous, some existence results of extremal solutions are obtained. Furthermore, we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.