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.     

对流扩散方程的基于加权本质非振荡插值的调整对流的修正特征差分解法

本文把J．Douglas提出的调整对流的修正特征差分法(MMOCAA,Numer．Math,1999,83：3553691和加权本质非振荡WENO插值相结合,提出了求解对流扩散方程的WENO-MMOCAA差分方法。此方法避免了原来基于高次(≥2)Lagrange插值的MMOCAA差分方法在解的大梯度附近所产生的震荡。本文给出了格式的误差估计及数值例子。     

一维非定常对流扩散方程非均匀网格上的高精度紧致差分格式

本文在非均匀网格上给出了求解非定常对流扩散方程的一种高精度紧致差分格式,特别适合边界层和大梯度等问题的求解.从稳态对流扩散方程入手,首先,基于非均匀网格上的泰勒级数展开对空间导数项进行离散,然后对时间项采用二阶向后欧拉差分公式,从而得到一维非定常对流扩散方程在非均匀网格上的三层全隐式紧致差分格式.新格式在时间具有二阶精度,空间具有三到四阶精度,并且是无条件稳定的.最后,通过数值实验验证了本文格式的精确性,以及在处理诸如边界层和大梯度问题上的优势.     

二维半导体问题的特征差分方法和L^∞模估计

采用了特征差分方法研究了二维半导体问题的电子和空穴浓度方程的非齐次牛曼问题，并用最大模原理得到Ｌ＾∞－模误差估计。与传统计算方法相比，特征差分方法有比较小的截断误差，且格式简单，可以对时间采用大步长计算。     

抛物型方程的一族高精度恒稳定的隐式差分格式

用待定参数法对一维抛物型方程构造了一族高精度恒稳定的隐式差分格式，格式的截断误差达O（Δt3 Δx6)，可用追赶法求解。     

求解变系数对流扩散反应方程的指数型高精度紧致差分方法

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法.我们首先将模型方程变形,借助常系数对流扩散方程的指数型高精度紧致差分格式,采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式;并从理论上分析了当Pelect数很大时,本文格式达到四阶计算精度时网格步长的限制条件;离散得到的代数方程组可采用追赶法直接求解.数值实验结果与理论分析完全吻合,表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点.     

改进的高阶加权紧致非线性差分格式

基于中心紧致三对角系数矩阵的四阶、六阶格式，通过非线性组合五阶WENO差分格式大模板和两个对称小模板对网格半节点函数值的插值计算，得到求解双曲守恒律方程的四阶、五阶加权紧致非线性差分格式。线性对流方程的计算结果验证了格式的计算精度和计算效率；一维无粘Burgers方程的计算结果验证了格式分辨率；一、二维欧拉方程的计算结果验证了格式对非线性问题中激波间断的捕捉能力。所有数值实验均表明，构造的新格式是一个高效、高精度、高分辨率的激波捕捉格式。     

KdV方程的一个紧致差分格式

本文基于经典的有限差分方法,讨论了满足周期边界条件的KdV方程的高精度差分格式的构造问题.通过引入中间函数及紧致方法对空间区域进行离散,提出了KdV方程的一个两层隐式紧致差分格式.利用泰勒展开法得出,该格式在时间方向具有二阶精度,但在空间方向可达到六阶精度.采用线性稳定性分析法证明了该格式是稳定的.数值结果表明:本文所提出的紧致差分格式是有效的,在空间方向拥有较高的精度,还能够很好地保持离散动量和能量守恒性质.     

空腔的流激振荡及其声激励抑制方法的数值模拟

通过数值模拟，研究了矩形空腔在亚音速外流下的流激振荡问题。采用显式ＭａｃＣｏｒｍａ－ｃｋ二步预估校正有限差分格式，求解二维雷诺平均非定常Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程；并用Ｃｅｂｅｃｉ－Ｓｍｉｔｈ代数紊流模型作适当修正来模拟紊流效应，对有、无前缘声激励两种情况的空腔流场作了数值模拟，计算的振荡频率及振荡幅值都与实验结果基本符合，并且较好地模拟出了声激励对流场的影响。     

对流占优扩散方程的改进特征差分算法

将特征线方法和有限差分方法相结合，给出了一种求解对流占优扩散方程数值解的新的隐式特征差分格式，并研究了新算法的收敛性，新算法的优点是适应性强，特别适用于变系数方程，数值试验的结果表明在消除数值震荡方面更有效。     

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.     

Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

It is known that Harten's uniformly non-oscillatory scheme is a second-order accurate scheme for discretizing coservation laws. In this paper a multigrid technique and Runge-Kutta time stepping with frozen dissipation is applied to Harten's scheme in order to obtain the steady-state solution. It is shown that these techniques applied to Harten's scheme lead to a better convergence to the steady-state solution of a first-order conservation law than applied to Jameson's scheme.     

求解浅水方程的五阶松弛格式(英文)

对浅水方程,提出了一种具有五阶精度的松弛格式。该格式以五阶WENO重构和隐式Runge-Kutta方法为基础。格式保持了松弛格式简单的优点,即不用求解Riemann问题和计算通量函数的雅可比矩阵。应用该方法对一维平底和非平底溃坝问题进行了数值模拟,结果表明方法健全、有效。对摩阻源项也进行了讨论。     

An analysis of the convection-diffusion problems using meshless and meshbased methods

The numerical solution of the convection-diffusion equation represents a very important issue in many numerical methods that need some artificial methods to obtain stable and accurate solutions. In this article, a meshless method based on the local Petrov-Galerkin method is applied to solve this equation. The essential boundary condition is enforced by the transformation method, and the MLS method is used for the interpolation schemes. The streamline upwind Petrov-Galerkin (SUPG) scheme is developed to employ on the present meshless method to overcome the influence of false diffusion. In order to validate the stability and accuracy of the present method, the model is used to solve two different cases and the results of the present method are compared with the results of the upwind scheme of the MLPG method and the high order upwind scheme (QUICK) of the finite volume method. The computational results show that fairly accurate solutions can be obtained for high Peclet number and the SUPG scheme can very well eliminate the influence of false diffusion.     

A time-based arc-length like method to remove step size effects during fracture propagation

An arc-length like method is presented which alters the size of the time increment when simulating crack propagation problems. By allowing the time increment to change during the time step a constraint can be imposed, which is used to enforce the fracture to propagate a single element length per time step. This removes the effect of the (interface) element size on propagating fractures, and therefore allows smooth fracture propagation during the simulation. The benefits of the scheme are demonstrated for three cases: mode-I crack propagation in a double cantilever beam, a shear fracture including inertial and viscoplastic effects in the surrounding material, and a pressurized fracture inside a poroelastic material. These cases highlight the ability of this scheme to obtain more accurate and nonoscillatory results for the force–displacement relation, to remove numerically induced stepwise fracture propagation, and to allow for arbitrary propagation velocities. An added benefit is that plastic strains surrounding a fracture are no longer affected by the (interface) element size.     

An unconditionally stable implicit method for hyperbolic conservation laws

Summary We construct a space-centered self-adjusting hybrid difference method for one-dimensional hyperbolic conservation laws. The method is linearly implicit and combines a newly developed minimum dispersion scheme of the first order with the recently developed second-order scheme of Lerat. The resulting method is unconditionally stable and unconditionally diagonally dominant in the linearized sense. The method has been developed for quasi-stationary problems, in which shocks play a dominant role. Numerical results for the unsteady Euler equations are presented. It is shown that the method is non-oscillatory, robust and accurate in several cases.     

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.     

A novel RBF-FD meshless scheme in curvilinear geometry for unbounded flows

A novel local radial basis function Radial Basis Function-Finite Difference (RBF-FD) scheme has been developed in curvilinear geometry and implemented to unbounded fluid flows. The far field boundary condition that arises due to the unboundedness of the fluid was handled efficiently and achieved higher order accurate results. The RBF-FD is combined with an upwind-based scheme to handle convective terms effectively. The effect of shape parameter on the accuracy of the results and the variation of shape parameter with the number of nodes are numerically investigated. The order of accuracy of the method is found in comparison with a finite difference scheme.     

计算平流过程的一种高精度离散格式及数值试验

该文提出了一种用泰勒-伽辽金有限元法进行数值模拟的高精度离散格式。应用此格式在旋转流场中对二维纯平流问题浓度场的变化进行了数值试验,并同迎风格式、Crank-Nicolson格式、Lax-Wendroff格式、蛙跳格式的数值模拟结果进行了对比分析。数值试验结果表明:该离散格式具有较高的精确度和较好的稳定性,还具有求解速度快、位相误差小的优点,适合应用于平流过程的高质量的模式中。