二维非线性对流扩散方程的特征混合有限元两重网格算法

本文构造了求解非线性对流扩散方程的两重网格算法,该算法首先是在步长为H的粗网格上求解一个非线性问题,再利用粗网格解得到一个线性问题并在细网格上求解一个线性问题.理论分析与数值计算表明,该算法不仅消除了数值振荡现象,还极大地提高了非线性对流扩散方程的计算效率.     

对流占优扩散方程的楔形基无网格法

传统的微分方程数值解方法求解对流占优扩散方程时,往往产生数值震荡现象,为了消除数值震荡,本文构建了一种新的数值求解方法――无网格方法进行数值求解。该方法采用配点法并引入一种新的楔形基函数构建了楔形基无网格方法,不需要网格划分,是一种真正的无网格方法,可以避免因为网格划分而影响计算效率。通过对新的楔形基函数的理论分析,证明了本文方法解的存在唯一性。最后,分别通过一维和二维的数值算例,表明该算法计算精度高,可以有效消除对流占优引起的数值震荡,是一种计算对流占优扩散方程数值解的高效方法。     

二维对流扩散方程的点插值算法

本文介绍了点插值原理,并针对二维对流扩散方程,构造带有多项式基的径向点插值无网格方法,对具体对算例沿特征线进行无网格计算,计算结果表明,新算法结构简单,且能取得很好的计算精度。     

定常对流扩散反应方程非均匀网格上高精度紧致差分格式

本文构造了非均匀网格上求解定常对流扩散反应方程的高精度紧致差分格式.我们首先基于非均匀网格上函数的泰勒级数展开,给出了一阶导数和二阶导数的高阶近似表达式;然后将模型方程变形,借助于对流扩散方程高精度紧致格式构造的方法,结合原模型方程,得到定常对流扩散反应方程的高精度紧致差分格式;最后给出的数值算例验证了本文格式高精度和高分辨率的优点.     

各向异性扩散问题的一个单元中心型有限体积格式

在辐射流体力学的数值模拟中,扩散算子的高效高精度离散是一个十分重要的问题.本文研究各向异性扩散方程在任意多边形网格上的数值求解问题,我们利用调和平均点和线性精确方法,构造了一个单元中心型有限体积格式.该格式只含有单元中心未知量,满足局部守恒条件,有紧凑的计算模板,在结构四边形网格上退化为一个九点格式.由于调和平均点插值算法是一个具有两点模板的二阶保正算法,因此,采用单元边上的调和平均点为插值节点,使得离散格式十分简洁,容易实施.此外,我们在格式构造中仅采用了二、三维网格的共有拓扑关系,使格式容易向三维问题推广,大部分程序代码可实现二、三维公用.我们采用典型的大变形扭曲网格及典型的扩散算例(包括连续和间断的扩散张量)对所提出的新格式进行了测试,数值算例表明,新格式在许多扭曲的多边形网格上具有二阶精度.     

对流-扩散-反应方程的变分多尺度解法

根据变分多尺度的思想求解了对流项和反应项占优的对流-扩散-反应方程.在变分多尺度思想的理论框架内,推导了附加于Galerkin变分弱形式的稳定化结构和具体的稳定化系数;阐述了这种稳定化结构和经典的SUPG稳定化结构之间的关系;数值算例表明,该稳定化系数可以适应均匀和非均匀的计算网格.通过网格的恰当加密,变分多尺度方法消除了算例中的数值伪振荡.     

求解对流扩散方程的一致高精度非振荡特征差分方法

把特征差分法和一致高精度非振荡插值相结合，提出了求解对流占优扩散问题的一致高精度非振荡特征差分格式，避免了标准的特征差分格式在陡峭前缘附近产生的伪振荡，给出了非线性差分格式的误差估计及数值算例。     

无网格彼得洛夫伽辽金法在大变形问题中的应用

将无网格局部彼得洛夫伽辽金(MLPG)法推广应用于大变形问题。导出了非线性局部子域对称弱形式,通过对该弱形式进行线性化得到了用于非线性计算的MLPG格式,并对MLPG的计算速度进行了优化,使MLPG成为一种复杂度为O(N)的算法。几何非线性和几何与材料双重非线性的数值算例表明,相对有限元方法,MLPG在处理此类大变形问题时收敛性好,精度高,并能减小有限元分析中易遇到的网格畸变带来的困难。     

对流扩散方程的特征线EFG算法

为了消除对流扩散方程因对流占优引起的数值震荡,本文首先将其转化为特征形式,并利用移动最小二乘基函数,构建了特征线无单元Galerkin方法．再对新建方法进行收敛性分析,分别给出关于支持域半径和时间步长的两种误差估计．最后,分别针对一维和二维算例进行了数值计算,并与有限元法进行了比较．数值结果表明,本文算法收敛性好,可以消除数值震荡,且通过选取合适的罚因子和支持域的无量纲尺寸,计算精度比有限元法更高,是求解对流占优扩散方程的一种有效程数值计算方法．     

求解二维Navier-Stokes方程的移动网格方法

为了减少解在较小的局部区域内有着很强的奇异性、剧烈变化等的偏微分方程求解问题的计算量,提出了一种基于方程求解的移动网格方法,并将其应用于二维不可压缩Navier-Stokes方程的求解．与已有的大部分移动网格方法不同,网格节点的移动距离是通过求解一个变系数扩散方程得到的,避免了做区域映射,也不需要对控制函数进行磨光处理,所以算法很容易编程实现．数值算例表明所提算法能够在解梯度较大的位置加密网格,从而在保证提高数值解的分辨率的前提下,可以很好地节省了计算量．由于Navier-Stokes 的典型性,所得算法能够推广到求解很大一类偏微分方程数值问题．     

Solution of Inverse Boundary Optimization Problem by Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm

The inverse boundary optimization problem, governed by the Helmholtz equation, is analyzed by the Trefftz method (TM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the inverse boundary optimization problem, the position for part of boundary with given boundary condition is unknown, and the position for the rest of boundary with additionally specified boundary conditions is given. Therefore, it is very difficult to handle the boundary optimization problem by any numerical scheme. In order to stably solve the boundary optimization problem, the TM, one kind of boundary-type meshless methods, is adopted in this study, since it can avoid the generation of mesh grid and numerical integration. In the boundary optimization problem governed by the Helmholtz equation, the numerical solution of TM is expressed as linear combination of the T-complete functions. When this problem is considered by TM, a system of nonlinear algebraic equations will be formed and solved by ECSHA which will converge exponentially. The evolutionary process of ECSHA can acquire the unknown coefficients in TM and the spatial position of the unknown boundary simultaneously. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. Besides, the stability of the proposed meshless method will be validated by adding some noise into the boundary conditions.     

Rosenau-Burgers方程一种新的数值解法

本文讨论Rosenau-Burgers方程初边值问题的数值解法.针对Rosenau-Burgers方程构造了一个新的差分格式,把网格分为奇、偶两套独立的网格,在偶数网格点采用显式格式,在奇数网格点采用Crank-Nicolson格式,这样偶、奇、显、隐交替的方法使计算量减少.同时针对非线性项进行了线性化,使格式的近似解...     

Free‐surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies

Several cases of nonlinear wave propagation are studied numerically in two dimensions within the framework of potential flow. The Laplace equation is solved with the harmonic polynomial cell (HPC) method, which is a field method with high‐order accuracy. In the HPC method, the computational domain is divided into overlapping cells. Within each cell, the velocity potential is represented by a sum of harmonic polynomials. Two different methods denoted as immersed boundary (IB) and multigrid (MG) are used to track the free surface. The former treats the free surface as an IB in a fixed Cartesian background grid, while the latter uses a free‐surface fitted grid that overlaps with a Cartesian background grid. The simulated cases include several nonlinear wave mechanisms, such as high steepness and shallow‐water effects. For one of the cases, a numerical scheme to suppress local wave breaking is introduced. Such scheme can serve as a practical mean to ensure numerical stability in simulations where local breaking is not significant for the result. For all the considered cases, both the IB and MG method generally give satisfactory agreement with known reference results. Although the two free‐surface tracking methods mostly have similar performance, some differences between them are pointed out. These include aspects related to modeling of particular physical problems as well as their computational efficiency when combined with the HPC method.     

A Fourth-Order Compact Finite Difference Scheme for Higher-Order PDE-Based Image Registration

Image registration is an ill-posed problem that has been studied widely in recent years. The so-called curvature-based image registration method is one of the most effective and well-known approaches, as it produces smooth solutions and allows an automatic rigid alignment. An important outstanding issue is the accurate and efficient numerical solution of the Euler-Lagrange system of two coupled nonlinear biharmonic equations, addressed in this article. We propose a fourth-order compact (FOC) finite difference scheme using a splitting operator on a 9-point stencil, and discuss how the resulting nonlinear discrete system can be solved efficiently by a nonlinear multi-grid (NMG) method. Thus after measuring the h-ellipticity of the nonlinear discrete operator involved by a local Fourier analysis (LFA), we show that our FOC finite difference method is amenable to multi-grid (MG) methods and an appropriate point-wise smoothing procedure. A high potential point-wise smoother using an outer-inner iteration method is shown to be effective by the LFA and numerical experiments. Real medical images are used to compare the accuracy and efficiency of our approach and the standard second-order central (SSOC) finite difference scheme in the same NMG framework. As expected for a higher-order finite difference scheme, the images generated by our FOC finite difference scheme prove significantly more accurate than those computed using the SSOC finite difference scheme. Our numerical results are consistent with the LFA analysis, and also demonstrate that the NMG method converges within a few steps.     

On comparing finite difference schemes applied to a nonlinear multi-region heat and mass transfer problem

 A three-time-level scheme is employed to obtain numerical solutions of heat and mass transfer potentials for the nonlinear Luikov system. Relative errors are examined when the numerical solutions are compared to the analytical solutions for a single layer linear problem using different finite difference techniques. As a final example, comparisons also are made when the numerical schemes are used to solve a two-layer multi-region nonlinear approach, considering that the thermal conductivity and mass conductivity of the material are dependent on temperature and moisture transfer potential respectively. Received 6 September 2000     

Numerical solutions of boundary detection problems using modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm

In this paper, the boundary detection problem, which is governed by the Laplace equation, is analyzed by the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA). In the boundary detection problem, the Cauchy data is given on part of the boundary and the Dirichlet boundary condition on the other part of the boundary, whose spatial position is unknown a priori. By adopting the MCTM, which is meshless and integral-free, the numerical solution is expressed by a linear combination of the T-complete functions of the Laplace equation. The use of a characteristic length in MCTM can stabilize the numerical procedure and ensure highly accurate solutions. Since the coefficients of MCTM and the position of part of the boundary are unknown, to collocate the boundary conditions will yield a system of nonlinear algebraic equations; the ECSHA, which is exponentially convergent, is adopted to solve the system of nonlinear algebraic equations. Several numerical examples are provided to demonstrate the ability and accuracy of the proposed meshless scheme. In addition, the consistency of the proposed scheme is validated by adding noise into the boundary conditions.     

Numerical solution of a parabolic system with coupled nonlinear boundary conditions

A numerical technique has been developed to solve a system that consists of m linear parabolic differential equations with coupled nonlinear boundary conditions. Such a system may represent chemical reactions, chemical lasers and diffusion problems. An implicit finite difference scheme is adopted to discretize the problem, and the resulting system of equations is solved by a novel technique that is a modification of the cyclic odd–even reduction and factorization (CORF) algorithm. At each time level, the system of equations is first reduced to m nonlinear algebraic equations that involve only the m unknown grid points on the nonlinear boundary. Newton's method is used to determine these m unknowns, and the corresponding Jacobian matrix can be computed and updated easily. After convergence is achieved, the remaining unknowns are solved directly. The efficiency of this technique is illustrated by the numerical computations of two examples previously solved by the cubic spline Galerkin method.     

An enthalpy formulation based on an arbitrarily deforming mesh for solution of the Stefan problem

Traditionally schemes for dealing with the Stefan phase change problem are separated into fixed grif or front tracking (deforming grid) schemes. A standard fixed grid scheme is to use an enthalpy formulation and track the movement of the phase front via a liquid fraction variable. In this paper, an enthalpy formulation is applied on a continuously deforming finite element grid. This approach results in a general numerical scheme that incorporates both front tracking and fixed grid schemes. It is shown how on appropriate setting of the grid velocity a fixed or deforming grid solution can be generated from the general scheme. In addition an approximate front tracking scheme is developed which can produce accurate non-oscillatory predictions at a computational cost close to an efficient fixed grid scheme. The versatility of the general scheme and the approximate front tracking scheme are demonstrated on solution of a number of Stefan problems in both one and two dimensions.     

材料非线性分析的自然单元法

自然单元法主要是基于给定结点的Voronoi图,利用自然相邻插值进行形函数的构造,其形函数满足Kronecker delta性质,便于施加本质边界条件,这使得自然单元法同时兼有有限单元法和无网格法的优点。在材料非线性本构关系的基础上,推导了考虑材料非线性问题的自然单元法模型。算例表明:该模型在处理材料非线性问题时,具有一定的合理性和可行性,是一种有效的数值方法。