一维定常型对流占优扩散方程的一类迎风有限体积格式

本文针对一维定常型对流占优扩散方程提出了一类迎风有限体积格式 .该格式对对流项具有二阶精度 ,对扩散项保持一阶精度 ,符合对流占优扩散问题强对流、弱扩散的特点 .     

空间-时间分数阶对流扩散方程的数值解法

本文考虑一个空间-时间分数阶对流扩散方程.这个方程是将一般的对流扩散方程中的时间一阶导数用α(0＜α＜1)阶导数代替,空间二阶导数用β(1＜β＜2)阶导数代替.本文提出了一个隐式差分格式,验证了这个格式是无条件稳定的,并证明了它的收敛性,其收敛阶为O(ι h).最后给出了数值例子.     

求解对流扩散方程的低耗散中心迎风格式

以四阶CWENO重构为基础,通过将对流项采用低耗散中心迎风格式离散,扩散项采用四阶中心差分格式离散,对得到的半离散格式采用四阶龙格库塔方法在时间方向上推进,得到一种求解对流扩散方程的高阶有限差分格式.数值结果验证了该格式的四阶精度和基本无振荡特性.     

对流扩散方程的四阶紧凑迎风差分格式

§1.引言 流动和传热传质的基本方程均是对流扩散型的.对流扩散方程的高阶紧凑差分格式,作为提高计算可靠性和节省计算量的一条有效途径,已引起相当的重视.作为该领域的一大进展,新近由Dennis推出的对流扩散方程四阶紧凑格式,在二维情形下呈九点式且勿须引入中间变量,只涉及对流扩散量本身,能在较粗网格下获取较为准确的数值结果.从本质上说,该格式系指数型四阶紧凑格式的多项式型翻版.它与指数型紧凑格     

变系数对流-扩散方程的交替分段Crank-Nicolson方法

对Saul'yev型格式中的对流项构造了一种新的离散化逼近形式,进而给出了变系数对流-扩散方程的Crank-Nicolson方法.这个方法是绝对稳定的.数值实验表明该方法并行性好,精度高,宜于直接在并行计算机上使用.     

一个半隐式指数型差分格式

为了用数值方法解对流-扩散方程,Allen-Southwell于1955年提出一种特殊形式的差分格式.这种格式与通常用差商代替微商所得到的差分方程不同,其系数带有指数函数,通常称此类差分格式为指数型格式.此后一直到1969年,苏联学者bH才首先证明了它对小参数的一致收敛性,使这类格式得到广泛的研究和应用.近几年来,许多人将隐式指数型格式用于解时间相关的对流-扩散方程,其最大缺点是:解多维     

求解对流扩散反应方程的四阶混合紧致差分方法

针对一维对流扩散反应方程,基于对流扩散方程的四阶指数型紧致差分格式,以及一阶导数的四阶Padé公式,发展了一种高效求解对流扩散反应方程的混合型四阶紧致差分格式.数值实验结果验证了格式对于边界层问题或大雷诺数或大Pelect数的大梯度问题的求解的高精度和鲁棒性的优点.     

扩展型动网格的Chebyshev有限谱方法

给出了基于非均匀网格的Chebyshev有限谱方法．提出了可生成两种类型扩展型动网格的均布格式．一种类型的网格被用来提高波面附近的分辨率,另一种类型则用在梯度较大的流动区域．由于采用Chebyshev多项式作为基函数,该方法具有高阶精度．从上个时间步到当前时间步,两套不均匀网格间的物理量采用Chebyshev多项式插值．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth预报格式和Adams-Moulton校正格式．为了避免由Korteweg-deVries(KdV)方程的弥散项引起的数值振荡,给出了一种非均匀网格下的数值稳定器．给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

求解非线性对流-扩散问题的特征—差分法

近年来,已有不少文献讨论发展型方程的特征-有限元或特征-差分解法.在这一方向上,J.Douglas,Jr.及T.F Russell[1]是重要的工作,它讨论了一维对流-扩散方程C(x)?u/?t+b(x)?u/?x-?/?x(a(x)?u/?x)=f(x,t)的数值求解问题,建立了基于线性插值与基于二次插值的两种特征-差分格式,给出了误差分析,将前一种格式推广到     

求解非线性对流-扩散问题的特征—差分法

近年来,已有不少文献讨论发展型方程的特征-有限元或特征-差分解法.在这一方向上,J.Douglas,Jr.及T.F Russell[1]是重要的工作,它讨论了一维对流-扩散方程C(x)?u/?t+b(x)?u/?x-?/?x(a(x)?u/?x)=f(x,t)的数值求解问题,建立了基于线性插值与基于二次插值的两种特征-差分格式,给出了误差分析,将前一种格式推广到     

Multiple domain dynamics simulated with coupled level sets

We adapt the level-set method to simulate epitaxial growth of thin films on a surface that consists of different reconstruction domains. Both the island boundaries and the boundaries of the reconstruction domains are described by different level-set functions. A formalism of coupled level-set functions that describe entirely different physical properties is introduced, where the velocity of each level-set function is determined by the value of the other level-set functions.     

Existence and uniqueness of solutions of surface reconstruction problem

Surface reconstruction from scattered data is an important problem in such areas as reverse engineering and computer aided design.In solving partial differential equations derived from surface reconstruction problems,level-set method has been successfully used.We present in this paper a theoretical analysis on the existence and uniqueness of the solution of a partial differential equation derived from a model of surface reconstruction using the level-set approach.We give the uniqueness analysis of the cl...     

Application of the Level-Set Model with Constraints in Image Segmentation

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive overrelaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.     

A hybrid level-set/moving-mesh interface tracking method

An approach for combining Arbitrary–Lagrangian–Eulerian (ALE) moving-mesh and level-set interface tracking methods is presented that allows the two methods to be used in different spatial regions and coupled across the region boundaries. The coupling allows interface shapes to be convected from the ALE method to the level-set method and vice-versa across the ALE/level-set boundary. The motivation for this is to allow high-order ALE methods to represent interface motion in regions where there is no topology change, and the level-set function to be used in regions where topology change occurs. The coupling method is based on the characteristic directions of information propagation and can be implemented in any geometrical configuration. In addition, an iterative method for the hybrid formulation has been developed that can be combined with pre-existing solution methods. Tests of a propagating interface in a uniform flow show that the hybrid approach provides accuracy equivalent to what one is able to obtain with either of the methods individually.     

A microscopic convexity theorem of level sets for solutions to elliptic equations

We study the microscopic level-set convexity theorem for elliptic equation Lu?=?f(x, u, Du), which generalize Korevaars?? result in (Korevaar, Commun Part Diff Eq 15(4):541?C556, 1990) by using different expression for the elementary symmetric functions of the principal curvatures of the level surface. We find out that the structure conditions on equation are as same as conditions in macroscopic level-set convexity results (see e.g. (Colesanti and Salani, Math Nachr 258:3?C15, 2003; Greco, Bound Value Prob 1?C15, 2006)). In a forthcoming paper, we use the same techniques to deal with Hessian type equations.     

Numerical Simulation of Dendritic Crystal Growth

We consider the modified Stefan-Problem with Gibbs-Thomson correction, but vanishing kinetic undercooling. In this case the interface velocity is not given by mean curvature flow, but has to be computed explicitly from the temperature gradients at the interface. A finite element method on adaptive refined multigrids is presented here. Dirichlet conditions have to be satisfied along the solid-liquid interface that in general may intersect the elements. We use an implicit level-set representation of the interface that preserves it as a sharp surface, in contrast to the phase-field method. In numerical simulations we observe dendritic patterns that show good agreement with different features of physical experiments. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A level-set method for shape optimization

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh. To cite this article: G. Allaire et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1125–1130.     

A level-set topological optimization method to analyze two-dimensional thermal problem using BEM

A level-set based topological optimization approach is proposed using boundary element method (BEM) to solve two-dimensional(2D) thermal problems. The objective function is considered as a function of temperature and thermal flux defined on boundaries with Dirichlet and Neumann boundary conditions. The topological sensitivity is derived combining BEM under the assumption of insulating topological boundaries generated during optimization. Smooth boundaries represented by the level-set function is updated using topological sensitivity with a regularization term. Numerical examples with different objective functions considering the real-world problems are presented to show the effectiveness of the proposed approach. The topological sensitivity, computational time and boundary smoothness are verified by comparing with finite difference method (FDM).     

Robust Contour Tracking Model Using a Variational Level-Set Algorithm

In this work, we provide a novel variational level-set based object contour tracking approach. Thus, a mathematically rigorous variant of the Chan-Vese algorithm for image segmentation via geometric active contour model is proposed here. With respect to the original contour detection algorithm, the level set function ?(t) defining the evolving contour S _t  = {x; ?(t, x) = 0} is iteratively computed from a nonlinear parabolic boundary value problem that is well posed in the space of functions with bounded variations. We provide a robust mathematical justification of the proposed level-set model.