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。     

全离散非线性Galerkin算法的有界性和稳定性

非线性Ｇａｌｅｒｋｉｎ算法是长时间范围内求解非线性发展方程的一种新的数值格式。我们在这篇文章里，提供了全离散非线性Ｇａｌｅｒｋｉｎ算法的有界性和稳定性结果。     

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...     

辐射扩散方程基于虚拟元方法的一个保正守恒格式

作为近年来广受关注的一种数值方法，虚拟元方法具有很多优势。但在求解实际问题导出的一些辐射扩散方程时，该方法可能无法保证数值解的非负性及一般多边形网格上的局部守恒性。针对辐射扩散方程，利用非线性两点流逼近方法作为后处理措施，提出了一种基于虚拟元方法的保正守恒格式。该格式通过最低阶虚拟元方法得到数值解的单元顶点值，再利用非线性两点流逼近方法得到数值解的非负单元中心值，同时使格式满足局部守恒性。任意多边形网格上的数值结果表明，该格式具有保正性和解的近似二阶收敛速度，对于处理含强间断或非线性扩散系数的辐射扩散问题均有较强的适应性。     

载流悬臂柱壳的非线性数值分析

针对在电磁场和机械场耦合作用下的载流柱壳的非线性变形问题的数值解法进行了研究。给出了载流柱壳在耦合场作用下的几何方程、物理方程、二维电动力学方程、磁弹性非线性运动方程和洛仑兹力表达式,建立了差分格式和线性化迭代方程,给出了这些方程的数值解系统,并以悬臂柱壳为例,计算了该壳在电磁场和机械载荷耦合作用下的应力及变形;讨论了其应力及变形与外加电磁参量之间的关系;证明了变化电磁参量可以对壳的工作状态实施控制。     

耦合非线性薛定谔方程的平均离散梯度法

能量守恒格式对于准确地模拟微分方程的运动具有重要的意义.本文应用平均离散梯度法和辛算法求解耦合非线性薛定谔方程.数值结果表明平均离散梯度法能很好地模拟耦合非线性薛定谔方程在不同参数下孤立波的演化行为,并能精确地保持方程的离散能量.平均离散梯度法比相应的辛格式更好地保持方程的能量守恒.     

几个非线性波动方程的数值方法

文采用一种线性隐格式来解Korteweg-de Vries(KDV)/(GKDV)方程,对这种方法做一下推广,就能应用到Kadomtsev-Petviashvili(KP)方程以及它的广义形式(GKP)方程。这种方法是无条件稳定的,且是无损耗的。数值实验描述了一个线性孤波运动的情形以及两个孤波交互的情形,从结果来看,它们满足孤立子解的两个守恒-动量守恒和能量守恒。     

一类非局部粘性水波模型的数值格式

本文主要讨论带非局部粘性项水波方程的数值方法.我们建立了一种求解这类粘性水波方程的数值方案.该方案有效解决了非局部粘性项与非线性项的离散问题.所提的格式包括对α阶分数阶项的2-α阶格式和对非线性项的线性化处理的混合格式.我们证明了这种格式是无条件稳定的,并得出线性Crank-Nicolson加2-α格式的收敛阶是O(?t32+N1-m)的结论.一系列的数值例子验证了理论证明的正确性.最后,我们用所提数值格式研究了粘性水波方程的渐近衰减率,并讨论了各种参数项对衰减率的影响.     

非线性Schr(o)dinger方程的显式模守恒格式

本文采用Magnus方法求解非线性Schrdinger方程。Schrdinger方程具有模平方守恒特性,用适当差分格式对其进行模平方守恒空间离散,转化成模平方守恒的常微分方程组,再用Magnus方法求解。数值结果表明Magnus方法能保非线性Schrdinger方程模守恒量的优越性和好的稳定性。Magnus方法可应用到其它模守恒的微分方程。     

非线性结构动力方程求解的显式差分格式的特性分析

本文针对软化非线性结构体系动力方程,分析了作者推导出的求解方程的显式差分格式的收敛性及稳定性,分别给出了结构体系处于非线性正刚度及屈服后的负刚度反应阶段时显式差分格式的稳定条件。稳定性分析结果表明,只要结构体系处于初始反应的粘-弹性阶段时显式差分格式满足稳定性条件,则可以保证软化非线性结构体系反应求解的整个过程中格式的计算稳定性。     

基于Leonard规正变量的修正高分辨率组合格式

通过对规正变量进行重构,本文提出了求解对流扩散方程的修正高分辨率组合格式,它能够求解边界层和大梯度等问题.首先,根据规正变量的定义得出了组合格式的通用表达式,然后对时间项采用二阶中心差分格式,得到了对流扩散方程的离散表达式,对离散化得到的代数方程组采用TDMA算法求解,并推导出了组合格式计算过程迭代收敛时所满足的充分条件.数值实验表明:新格式具有分辨率高,数值耗散较低,总偏差量较小,能很好模拟场变量的大梯度变化,计算结果优于传统格式.     

Crank-Nicolson Quasi-Wavelet Based Numerical Method for Volterra Integro-Differential Equations on Unbounded Spatial Domains

The numerical solution of a parabolic Volterra integro-differential equation with a memory term on a one-dimensional unbounded spatial domain is considered. A quasi-wavelet based numerical method is proposed to handle the spatial discretisation, the Crank-Nicolson scheme is used for the time discretisation, and second-order quadrature to approximate the integral term. Some numerical examples are presented to illustrate the efficiency and accuracy of this approach.     

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.     

Navier-Stokes/Darcy 模型的 BDF2 模块化梯度散度稳定格式的数值分析

Navier-Stokes/Darcy 方程可用来模拟河流中的污染物对地下水的污染问题,以及血液在血管及器官间的渗透问题等,由于其在实际中的广泛应用,对其数值方法的研究受到广泛关注。提出了求解Navier-Stokes/Darcy方程的BDF2模块化梯度散度稳定数值格式,这种格式通过增加稳定化项,提高了解的有效性和精确性,在保留梯度散度稳定格式优点的同时,可以有效的避免大的稳定化参数对解的非正常影响,给出了格式的稳定性和误差分析。最后,通过数值算例验证了理论分析的正确性。     

Numerical analysis of the steady-state behaviour of a MOS field effect transistor

A numerical scheme is developed to simulate the non-isothermal steady-state behaviour of a MOS field effect transistor. In a desire to develop a fast, stable numerical scheme, physical instabilities were eliminated by using a simplified device model. The numerical technique developed permits a computer solution of the majority carrier transport equation, the nonlinear heat conduction equation, in which the heat generation term is obtained from the solution of the transport equation, and a number of auxiliary differential equations. The simplified model of the MOS transistor adopted will not, of course, produce any information on the actual operation of the short channel MOS transistor of practical interest today, but the numerical scheme can be extended to simulate short channel models that are of great practical interest.     

An efficient semi-implicit finite element scheme for two-dimensional tidal flow computations

A semi-implicit finite element scheme is proposed for two-dimensional tidal flow computations. In the scheme, each term of the governing equations, rather than each dependent variable, is expanded in terms of the unknown nodal values and it helps to reduce computer execution time. The friction terms are represented semi-implicitly to improve stability, but this requires no additional computational effort. Test cases where analytic solutions have been obtained for the shallow water equations are employed to test the proposed scheme and the test results show that the scheme is efficient and stable. An numerical experiment is also included to compare the proposed scheme with another finite element scheme employing Serendipity-type Hermitian cubic basis functions. A numerical model of an actual bay is constructed based on the proposed scheme and computed tidal flows bear close resemblance to flows measured in field survey.     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

算法的有界性和稳定性(英文)

非线性Ｇａｌｅｒｋｉｎ算法是长时间范围内求解非线性发展方程的一种新的数值格式。我们在这篇文章里,提供了全离散非线性Ｇａｌｅｒｋｉｎ算法的有界性和稳定性结果。     

任意多边形上非定常扩散方程的单元中心型有限体积格式

针对二维非定常扩散方程,构造适用于任意多边形网格的单元中心型有限体积格式。采用向后欧拉格式进行时间离散,空间上在离散扩散算子时,利用网格顶点作为辅助插值点,通过求解一个欠定方程组将辅助插值点信息替换成网格单元中心点信息,最终得到只含单元中心未知量的离散格式。该格式既满足局部守恒条件,又满足线性精确准则。在几类多边形网格上进行数值实验,分别考虑扩散系数是连续和间断的情况,发现新格式均可达到二阶收敛。其数值表现显著优于算数平均加权和逆距离加权的九点格式,与双线性插值的加权方式结果相近,并且克服了双线性插值加权方式不适用于三角形网格的弊端。数值算例表明新格式求解非线性扩散方程仍然可以达到二阶收敛。