计算半线性椭圆问题多解的一类谱Galerkin型搜索延拓法的收敛性分析

本文提出计算半线性椭圆边值问题多解的一类高效的谱Galerkin型搜索延拓法(SGSEM).该方法基于模型方程相应线性特征值问题的若干特征函数的线性组合构造多解初值,充分利用了传统搜索延拓法构造多解初值方面的优势.同时,采用插值系数Legendre-Galerkin谱方法离散模型问题,具有计算成本低、计算精度高的优点.运用Schauder不动点定理和其他技巧,本文严格证明了对应于每个特定真解的数值解的存在性以及限制在该真解一个充分小的邻域内的数值解的唯一性,并证明了其谱收敛性.数值结果验证了算法的可行性与高效性,并展示了不同类型的多解.     

一类奇异抛物方程淬火解的数值分析

本文利用有限差分方法研究一类带奇异Neumann边界条件和奇异反应项的半线性抛物方程数值解的渐近行为.在初值满足一定假设的条件下,证明了数值解淬火速率与连续解淬火速率的一致性,数值淬火时间收敛于连续淬火时间,并通过数值实验验证了理论分析.     

带奇性的非线性抛物方程边值问题

本文讨论非线性奇异抛物方程第一边值问题解的存在性、唯一性、稳定性以及当t充分大时解的渐近性态．利用先验估计的方法得到：存在唯一的光滑正解，解在L^1范数意义下连续依赖于初值．t充分大时，||u-u↑-||L^2收敛于一个常数．     

Hamilton-Jacobi方程的小波Galerkin方法

本文选择Daubechies小波尺度函数空间作为Galerkin方法的测试函数空间,并将其应用于Hamilton-Jacobi方程,得到了求解Hamilton-Jacobi方程的小波Galerkin方法的数值格式．由于小波在时间和频率上的局部性,本方法适用于处理具有奇异解的问题,可以有效地防止数值振荡．数值试验显示,本方法是有效的．     

利用变分迭代技术解时滞微分方程

应用变分迭代法这种较新的迭代技术解具有初值条件的时滞微分方程.通过这种方法,获得了它的数值解和精确解.通过一些实例充分地说明了这种方法是有效地和便捷的,所得的值与精确解相比较,进一步表明了这种方法的可靠性和精确性.而且这种方法还能被应用到其它领域.     

热传导方程的小波解法

本文利用微分算子的小波表示，讨论一维热传导方程初值问题的Daubechies小波解，给出此问题的显式离散格式。由于小波在时间和频率上的局部性，此方法特别适用于有奇异解的热传导方程，逼近精度高，而且没有发生解的振荡现象。     

不可压液晶模型经典解的存在性（英文）

本文研究一个描述离子在向列型液晶中输运和扩散的非线性偏微分方程模型.该模型耦合了对应于电势满足Maxwell’s方程的离子的连续性方程的Nernst-Planck系统,控制液晶流演变的不可压Naiver-Stokes方程与关于液晶方向场的非线性Allen-Cahn型方程.我们利用能量方法证明了该系统的大初值经典解的局部存在性和小初值经典解的整体存在性.     

磁砸箍模型全局强解的存在性

磁砸箍是现代等离子体研究领域中的一个重要模型,本文研究磁砸箍的初边值问题,利用一些精细的全局先验估计,证明具有轴对称初值的磁砸箍模型全局强解的存在性.这些初值可以充分大并且包含真空.     

非协调曲边四边形十二自由度平板弯曲单元*

本文利用精确元法[1],给出一个十二自由度曲边四边形板弯曲单元.该方法不需要变分原理,适用于任意正定和非正定偏微分方程.利用这个方法,单元之间的协调条件很容易满足,仅须位移和内力在单元节点上连续,即可保证所得到的解收敛于精确解.利用本文方法所获得的解,无论是位移还是内力可同时有二阶收敛精度.文末给出数值算例.表明了本文所得到的单元有非常好的精度.     

二维刚性边值问题的小波-吉尔解法

本文利用微分方程数值解的离散小波表示，讨论了此类方程在满足一定初始条件和边值条件下，在一个方向上利用小波伽辽金法，另一方向上利用吉尔方法进行求解，提出了一种解二维刚性初，边值问题的小波数值算法，计算结果表明，利用该方法所求得的数值解精度高，而且由小波特有的性质，它特别适用于求解带有奇异摄动的刚性问题。     

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

This paper is devoted to the testing and comparison of numerical solutions obtained from higher-order accurate finite difference schemes for the two-dimensional Burgers' equation having moderate to severe internal gradients. The fourth-order accurate two-point compact scheme, and the fourth-order accurate Du Fort Frankel scheme are derived. The numerical stability and convergence are presented. The cases of shock waves of severe gradient are solved and checked with the fourth-order accurate Du Fort Frankel scheme solutions. The present study shows that the fourth-order two-point compact scheme is highly stable and efficient in comparison with the fourth-order accurate Du Fort Frankel scheme.     

A composite semi-conservative scheme for hyperbolic conservation laws

In this work a first order accurate semi-conservative composite scheme is presented for hyperbolic conservation laws. The idea is to consider the non-conservative form of conservation law and utilize the explicit wave propagation direction to construct semi-conservative upwind scheme. This method captures the shock waves exactly with less numerical dissipation but generates unphysical rarefaction shocks in case of expansion waves with sonic points. It shows less dissipative nature of constructed scheme. In order to overcome it, we use the strategy of composite schemes. A very simple criteria based on wave speed direction is given to decide the iterations. The proposed method is applied to a variety of test problems and numerical results show accurate shock capturing and higher resolution for rarefaction fan.     

Numerical implementation of seismic wavefield operators via path integrals

The seismic imaging problem centers around mathematical and numerical techniques to create an accurate image of the earth's subsurface, using recorded data from geophones that capture reflected seismic waves. Using a path integral approach, a wavefield extrapolater can be expressed as a limit of depth-sliced path steps through a variable velocity medium. An image is created from the correlation between upward and downward going waves. We report on the mathematical issues that arise in implementing numerical algorithms based on the path integral approach, in particular convergence, stability, and accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axially symmetric wave motion of a viscous liquid in a linearly viscoelastic tube

The problem of the propagation of progressive waves in a tube made of a linearly viscoelastic material which encloses a viscous Newtonian liquid is examined. For numerical calculations, it is proposed that the behavior of the tube wall material be described by the Voigt model. Dispersion curves are constructed for this case.S. M. Kirov Azerbaidzhan State University, Baku. Translated from Mekhanika Polimerov, No. 2, pp. 317–321, March–April, 1976.     

Plane wave decomposition in the unit disc: Convergence estimates and computational aspects

This paper deals with the numerical simulation of time-harmonic wave fields using progressive plane waves. It is shown that a plane wave travelling in arbitrary direction can be numerically recovered with an accuracy of the order of the machine precision with a collocation formulation and the square root of the machine precision with a least-square formulation. However, strongly evanescent and nearly singular wave fields cannot be properly recovered with standard double-precision floating-point arithmetic. Some of the ideas are applied to the elastic wave equation and a simple optimization algorithm is proposed to find a good compromise between the accuracy and the number of plane waves.     

On a kinetic model for shallow water waves

The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter ? which leads for ? → 0 to the system of shallow water waves. The corresponding ‘equilibrium’ distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.     

Application of the Chebyshev pseudospectral method to van der Waals fluids

In this paper, we consider a class of van der Waals flows with non-convex flux functions. In these flows, nonclassical under-compressive shock waves can develop. Such waves, which are characterized by kinetic functions, violate classical entropy conditions. We propose to use a Chebyshev pseudospectral method for solving the governing equations. A comparison of the results of this method with very high-order (up to 10th-order accurate) finite difference schemes is presented, which shows that the proposed method leads to a lower level of numerical oscillations than other high-order finite difference schemes and also does not exhibit fast-traveling packages of short waves which are usually observed in high-order finite difference methods. The proposed method can thus successfully capture various complex regimes of waves and phase transitions in both elliptic and hyperbolic regimes.     

Solitary wave solutions of the modified equal width wave equation

In this paper we use a linearized numerical scheme based on finite difference method to obtain solitary wave solutions of the one-dimensional modified equal width (MEW) equation. Two test problems including the motion of a single solitary wave and the interaction of two solitary waves are solved to demonstrate the efficiency of the proposed numerical scheme. The obtained results show that the proposed scheme is an accurate and efficient numerical technique in the case of small space and time steps. A stability analysis of the scheme is also investigated.     

Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation

Summary. Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equation For p>5 , the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics. Received March 28, 2000; accepted August 24, 2000 %%Online publication November 15, 2000 Communicated by Thanasis Fokas     

A collocation method with cubic B-splines for solving the MRLW equation

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.