Waveform Relaxation Methods for Lie-Group Equations

In this paper, we derive and analyse waveform relaxation (WR) methods for solving differential equations evolving on a Lie-group. We present both continuous-time and discrete-time WR methods and study their convergence properties. In the discrete-time case, the novel methods are constructed by combining WR methods with Runge-Kutta-Munthe-Kaas (RK-MK) methods. The obtained methods have both advantages of WR methods and RK-MK methods, which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold. Three numerical experiments are given to illustrate the feasibility of the new WR methods.     

中立抛物型偏微分方程解的强迫振动

本文讨论一类多滞量中立抛物型偏微分方程解的振动性质 ,获得了其一切解振动的充分条件 ;指出了与普通抛物型偏微分方程质的差异 .     

Charpit System of Partial Differential Equations with Algebraic Constraints

In this paper the Charpit system of partial differential equations with algebraic constraints is considered. So, first the compatibility conditions of a system of algebraic equations and also of the Charpit system of partial differential equations are separately considered. For the combined system of equations of both types sufficient conditions for the existence of a solution are found. They lead to an algorithm for reducing the combined system to a Charpit system of partial differential equations of dimension less than the initial system and without algebraic constraints. Moreover, it is proved that this system identically satisfies the compatibility conditions if so does the initial system.     

脉冲时滞抛物型偏微分方程组解的振动性

研究一类脉冲时滞抛物型偏微分方程组解的振动性,利用一阶脉冲时滞微分不等式获得了该类方程组在两类不同边值条件下所有解振动的若干充分条件.所得结果充分反映了脉冲和时滞在振动中的影响作用.     

脉冲中向量中立型抛物偏微分方程的H-振动性

研究一类脉冲向量中立型抛物偏微分方程的振动性,借助Domslak引进的H-振动的概念及内积降维的方法,将多维振动问题化为一维脉冲中立型微分不等式不存在最终正解的问题,建立了该类方程在Dirichlet边值条件下所有解H-振动的若干充分判据,这里H是R~M中的单位向量.     

Multigrid for second‐order ADN elliptic systems

This paper theoretically examines a multigrid strategy for solving systems of elliptic partial differential equations (PDEs) introduced in the work of Lee. Unlike most multigrid solvers that are constructed directly from the whole system operator, this strategy builds the solver using a factorization of the system operator. This factorization is composed of an algebraic coupling term and a diagonal (decoupled) differential operator. Exploiting the factorization, this approach can produce decoupled systems on the coarse levels. The corresponding coarse‐grid operators are in fact the Galerkin variational coarsening of the diagonal differential operator. Thus, rather than performing delicate coarse‐grid selection and interpolation weight procedures on the original strongly coupled system as often done, these procedures are isolated to the diagonal differential operator. To establish the theoretical results, however, we assume that these systems of PDEs are elliptic in the Agmon–Douglis–Nirenberg (ADN) sense and apply the factorization and multigrid only to the principal part of the system of PDEs. Two‐grid error bounds are established for the iteration applied to the complete system of PDEs. Numerical results are presented to illustrate the effectiveness of this strategy and to expose factors that affect the convergence of the methods derived from this strategy.     

线性随机微分方程的离散波形松弛方法的几乎必然收敛性

提出了随机微分方程的离散型波形松弛方法,证明了它是几乎必然收敛的.此外,通过数值实验验证了所得结果.     

偏微分方程的调和分析方法简介

本文致力于阐述调和分析与现代偏微分方程研究的关系,特别是奇异积分算子、拟微分算子、Fourier限制性估计、Fourier频率分解方法在椭圆边值问题、非线性发展方程研究中的重要作用.对于偏微分方程研究的各种方法进行了比较与分析,指出了偏微分方程的调和分析方法的优点与局限性.与此同时,还给出了偏微分方程的调和分析方法这一领域的最新研究进展.     

求解Oseen流的交替线松弛多重网格方法

利用Riemann解的通量差分分裂法——Godunov方法对Oseen流控制方程进行离散,得到了基于一阶上迎风格式的离散方程,并给出了使用多重网格方法求解该离散方程的V-循环算法和W-循环算法的收敛性分析.通过局部Fourier分析方法,对获得的离散方程的聚对称交替线GaussSeidel松弛的光滑性质进行了研究.结果表明:使用多重网格的两层网格及三层网格算法求解具有不同Reynolds数的Oseen流,即便是在高Reynolds数情况下,聚对称交替线Gauss-Seidel松弛具有很好的光滑性质,多重网格W-循环算法收敛性比V-循环算法好.     

抛物泛函微分方程解的振动

该文讨论一类抛物泛函微分方程解的振动性质，得到了不同边界条件下方程解振动的充分条件，并给出了实例．     

Stochastic Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

一类求解反应扩散方程的Newton波形松弛方法

对反应扩散方程提出一种新型的Newton波形松弛方法,并给出此方法的误差估计式.通过与传统的波形松弛方法比较,这种Newton波形松弛方法有更快的收敛性,且收敛速度不随网格加密而减慢.这种方法可以保持传统波形松弛方法可并行的特点.最后通过数值算例验证这种方法的有效性.     

Global Meromorphic Solutions of Partial Differential Equations

In this paper, we survey some recent progress on analytic theory of partial differential equations.     

Weak Continuity and Compactness for Nonlinear Partial Differential Equations

This paper presents several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. The compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws are first analyzed, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropic flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multi-dimensional steady compressible fluids. Then the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry are revealed. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.     

随机偏微分方程——建模,分析与有效动力学

综述随机偏微分方程的基本概念、理论、方法与应用,内容包括Hilbert空间中的Wiener过程、Ito随机积分、随机偏微分方程的解及其有效动力学。还介绍了随机偏微分方程的粗糙轨道、正则结构以及在Kardar-ParisiZhang(KPZ)方程中的应用。还介绍了段金桥与王伟的著作《Effective Dynamics of Stochastic Partial Differential Equations(随机偏微分方程的有效动力学)》的基本内容。     

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDEs

偶数阶拟线性中立型偏微分方程系统的振动性定理

获得了一类偶数阶拟线性偏泛函微分方程系统振动的若干充分条件.     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）在ｕ０有界可测，ｆ连续且ｆ＞０条件下，其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０，ｕ（０，ｘ）＝ｕ０（ｘ）的熵解，即论证了“沾性消失法”解此方程的正确性，１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.