一类交错网格的Gauss型格式

本文在交错网格的情况下 ,利用 Gauss型求积公式构造了一类不需解 Riemann问题的求解一维单个双曲守恒律的二阶显式 Gauss型差分格式 ,证明了该格式在CFL条件限制下为 TVD格式 ,并证明了这类格式的收敛性 ,然后将格式推广到方程组的情形 .由于在交错网格的情况下构造的这类差分格式 ,不需要求解 Riemann问题 ,因此这类格式与诸如 Harten等的 TVD格式相比具有如下优点 :由于不需要完整的特征向量系 ,因此可用于求解弱双曲方程组 ,计算更快、编程更加简便等 .     

一个求解Euler方程的特殊矩阵分裂格式

§1.引言 自[1]提出矢通量分裂格式以来,在求解气动方程方面得到广泛应用。矢通量分裂格式是一种求解守恒型双曲方程组的方法,它将方程中代表质量、动量和能量的矢通量按照矢通量Jacobian矩阵正负特征值分裂为两个亚矢通量项,目的在于改进显式格式和隐式格式的计算效率和提高求解时的稳定性。在求解方法上,对于二维问题,需要求解以4×4块矩阵为矩阵元的上三角矩阵和下三角矩阵,比中心差分格式需要求解两个块三     

用非精确的平行割线法求解奇异问题

奇异方程经常出现在很多实际非线性问题中,如反应扩散系统等.因此,研究奇异非线性方程的求解具有十分重要的意义.平行割线法是一种经典的求解非线性方程的迭代方法,它收敛阶较高,计算量较少.但在解决实际问题时,一方面,抽象出的数学模型与实际问题总是存在着一定的偏差,另外,在数据的计算中难免存在着一定的计算误差,所以研究用非精确的平行割线法求解非线性奇异问题具有很重要的现实意义,使得求解奇异问题具有更高的实用性和可行性.采用在平行割线法的迭代公式中加入摄动项的方法,构造出新的加速迭代格式,证明了新的迭代格式的收敛性,给出了收敛速率,得到了误差估计.     

求解非线性方程七阶收敛的牛顿迭代修正格式

本文研究了非线性方程求解的问题.利用泰勒公式和耦合方法,获得了一种求解非线性方程的加速收敛的七阶迭代改进格式,该格式不需要计算高阶导数,且具有更大的收敛半径,大大提高了计算效率.     

求解非线性方程七阶收敛的牛顿迭代修正格式（英文）

本文研究了非线性方程求解的问题.利用泰勒公式和耦合方法,获得了一种求解非线性方程的加速收敛的七阶迭代改进格式,该格式不需要计算高阶导数,且具有更大的收敛半径,大大提高了计算效率.     

求函数稳定点的反插值算法及其收敛速率

§1.引言 一维搜索在非线性规划中非常重要,它常可归结为方程f′(x)=0的求解问题.本文基于牛顿反插值法对该问题提出了一个迭代求解格式,对于一般的n点迭代格式,该算法利用前n点的信息构造迭代的第n+1点.因此具有良好的局部收敛性;而且计算格式简单,易于计算机实现.数值试验表明,用三点格式已收敛得很快.     

一种抑制激波计算中数值振荡现象的双重小波收缩方法

在激波数值计算中,容易出现数值振荡的问题,振荡激烈时会掩盖真实解,为此提出了许多高精度复杂计算格式或采用人工粘性抑制数值振荡.从信号处理的角度,提出双重小波收缩方法,它能自适应提取激波数值振荡解中的真实物理解.先用局部微分求积法求解浅水波方程和理想流体Euler运动方程中的激波问题,发现其数值振荡现象严重,然后采用双重小波收缩方法对其处理,获得了无数值振荡解,它能准确捕捉激波的位置并且保持激波结构.相比于复杂的Riemann(黎曼)求解格式,借助小波收缩方法,可以采用相对简单的计算格式如微分求积法求解激波问题.     

中心格式在Saint-Venant方程组上的应用研究

浅水波方程组对于其数值格式有较高的要求.在实际应用中作者更关心在稳态解附近的行为,特别是当计算区域出现干湿界面的时候,不但要求格式具有和谐性,而且需要保持水深恒为非负,同时又要求数值格式具有较高的精度.设计同时满足这些性质的数值格式具有一定的难度.论文的核心是总结研究了受关注的求解浅水波方程组的中心格式:KP格式,BCKN格式和T格式的各自优势以及不足之处.该文通过求解一维问题来展示各自格式在一些算例上的应用.     

扩散问题的一类带有加权系数的隐格式及多子域并行算法

针对扩散问题提出了一类带有加权系数的隐格式,采用分组显式和区域分解思想,又构造了若干分组显式格式.结合初边值条件,建立了求解扩散问题的一种多子域并行算法.虽然格式是隐式的,但在算法实现过程中可显式且并行地计算,这样避免了求解线性方程组的复杂性.并且当加权系数1≤θ≤2.4时,格式是无条件稳定的;0θ1时,趋向于1的方向,格式也是无条件稳定的;θ=2时,算法收敛的最快,收敛速率接近于2.通过数值试验证明此类隐格式和并行算法是有效的,计算速度快,精确度高,易于实现并行.     

小波伽辽金有限元法在梁板结构中的应用

本文给出了基于小波尺度函数展开的高阶导数及其在伽辽金有限元法中有关联的导数乘积积分的计算格式，从而实现了将小波伽辽金法用于求解高于二阶导数微分方程边值问题的数值计算，使其在结构力学问题求解中成为可能·数值算例表明：本方法具有良好的计算精度·     

On the bidomain problem with FitzHugh–Nagumo transport

The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.     

The Relation and Evolution of Squeezing and Instability for Systems with Quadratic Hamiltonians

We propose a method that allows relating the quantum squeezing effect to the classical instability by establishing evolution equations for elements of the dispersion matrix directly in terms of elements of the stability matrix. The solution of these equations is written in terms of the evolution operator. Knowing this operator, we can analyze the system instability at finite times. Based on the developed formalism, we investigate two physical systems: the degenerate and nondegenerate parametric amplifiers with external -shaped pulses. We show that we can either amplify or, on the contrary, weaken both the squeezing effect and the system instability using -pulses.     

MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHRODINGER EQUATIONS WITH WAVE OPERATOR

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.     

On one approach to a posteriori error estimates for evolution problems solved by the method of lines

Summary. In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the -norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented. Received September 2, 1999 / Revised version received March 6, 2000 / Published online March 20, 2001     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

Numerical conformal mapping based on the generalised conjugation operator

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators.

     

A nonlinear Hamiltonian structure for the Euler equations

The Euler equations for inviscid incompressible fluid flow have a Hamiltonian structure in Eulerian coordinates, the Hamiltonian operator, though, depending on the vorticity. Conservation laws arise from two sources. One parameter symmetry groups, which are completely classified, yield the invariance of energy and linear and angular momenta. Degeneracies of the Hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point-wise conservation of vorticity. It is conjectured that no further conservation laws exist, indicating that the Euler equations are not completely integrable, in particular, do not have soliton-like solutions.     

On resolvent conditions and stability estimates

As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.     

Additive Difference Schemes and Iteration Methods for Problems of Mathematical Physics

We construct additive difference schemes for first-order differential–operator equations. The exposition is based on the general theory of stability for operator–difference schemes in lattice Hilbert spaces. The main focus is on the case of additive decomposition with an arbitrary number of mutually noncommuting operator terms. Additive schemes for second-order evolution equations are considered in the same way.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.