基于Gauss全局径向基函数的近岸浅水变形波高数值计算新方法

应用Gauss全局径向基函数来模拟波浪浅水变形波高变化方程中的未知函数,经实例分析探讨得到了一种可用于求解该方程数值解的新方法,并将其计算结果与常用数值分析方法得到的数值解相互对比印证,证明了基于Gauss全局径向基函数法计算结果的正确性.经验证,Gauss径向基函数法的平均计算误差相比其他方法均要小,表明该方法拥有更高的计算精度.同时,根据Gauss全局径向基函数的逼近结果,得出了浅水变形波高变化微分方程数值解的拟合函数,在实际工程中,可以利用该拟合函数来代替原方程的解析解,研究成果可为求解近岸浅水区域波浪运动提供一种新思路.     

用格子Boltzmann方法模拟非线性热传导方程

对具有非线性源项和非线性扩散项的热传导方程建立格子Boltzmann求解模型.在演化方程中增加了两个关于源项分布函数的微分算子,对演化方程实施Chapman-Enskog展开.通过对演化方程的进一步改进,恢复出具有高阶截断误差的宏观方程.对不同参数选取下的非线性热传导方程进行了数值模拟,数值解与精确解吻合得很好.该模型也可以用于同类型的其他偏微分方程的数值计算中.     

基于线性化Navier-Stokes方程二维水槽内单涡到双涡的数值模拟

建立了Navier-Stokes方程的预估-校正有限差分方法,在此基础上求得了二维水槽内部单涡到双涡的数值解,所得结果与前人的数值结果和解析解吻合很好.数值模拟结果表明,自由振动运动中自由面波高因粘性作用会发生衰减,且Reynolds数越大衰减越缓慢.在短时间内倾斜加速度激励下对于不同Reynolds数会出现一定周期的单涡.经过长时间的倾斜激励,水槽内涡场由单涡变化成双涡,而且只在较低的Reynolds数条件下出现双涡.     

带Markov状态转换的跳扩散方程的数值解

研究了一类带Markov状态转换的跳扩散方程的数值解的问题,为讨论这类方程精确解的数值计算问题,我们给出了一种基于Euler格式的方程解的跳适应算法,并在一定的条件下,证明了基于这种新的跳适应算法所得到的方程的数值解是收敛于它的精确解,同时还给出了数值解收敛到其精确解的收敛阶数.最后,本文通过两个例子说明了这种跳适应算法的计算有效性.     

带移动奇异源热方程的移动网格方法

研究了一个带若干奇异源热方程的数值求解,其源的移动由一个常微分方程描述.基于移动观察区域和区域分解思想提出了一个移动网格预估校正算法.网格方程可自然的通过并行高效求解,算法避免了跳跃信息[u]的计算而使物理方程的离散格式变得非常简单,且仍保持了空间上的二阶收敛性.数值例子验证了算法的收敛性和高效性,并模拟了非线性源函数带来的爆破现象.     

广义带导数的非线性Schrdinger方程的动态分析和精确解

利用动力系统方法,针对广义带导数的非线性Schrdinger方程的精确解问题进行研究分析.采用行波变换,将其化为常微分方程动力系统;计算出该方程动力系统的首次积分,讨论了系统在不同参数条件下的奇点与相图,得到对应的精确解,包括孤立波解、周期波解、扭结波解和反扭结波解.运用数值模拟的方法,对方程的光滑孤立波解和周期波解等进行了数值模拟.分析计算获得的结果完善了相关文献已有的研究成果.     

Euler型梁-柱结构的非线性稳定性和后屈曲分析

在有限变形的假设下,建立了位于非线性弹性基础上非线性弹性Euler型梁-柱结构的广义Hamilton变分原理,并由此导出了任意变截面Euler型梁-柱结构的3维非线性数学模型,其中考虑了转动惯性、几何非线性、材料非线性等因素的影响．作为模型的应用,分析了弹性基础上一端完全固支另一端部分固支,并受轴力作用的均质等截面线性弹性Euler型梁的非线性稳定性和后屈曲;结合打靶法和Newton法,给出了一种计算平凡解(前屈曲状态)、分叉点(临界载荷)和分叉解(后屈曲状态)的数值方法,对前两个分支点和相应分支解,成功地实现了数值计算,并考虑了基础反力和惯性矩对分支点的影响．     

二维非线性Schr(o)dinger方程的辛与多辛格式

对满足周期边界条件的二维非线性Schrödinger方程,运用中心差分对该方程进行空间离散, 得到一个有限维Hamilton系统,然后用隐式Euler中点格式进行时间离散得到其辛格式. 针对该方程的多辛形式, 运用有限体积法离散,得到一种直平行六面体上的中点型多辛格式. 用所构造的辛与多辛格式对二维非线性Schrödinger方程的平面波解和奇异解进行数值模拟,结果验证了所构 造格式的有效性. 最后, 根据计算结果,对两种格式进行了分析和比较.       

电池系统建模中Butler-Volmer方程的同伦分析求解

Butler-Volmer方程是电化学系统中描述电极动力学过程的本构方程,具有强非线性．为了对这一方程(耦合两个Ohm方程)进行解析求解,在同伦分析方法的框架下,发展了满足简单条件的广义非线性算子的算法,以取代原同伦分析中的非线性算子．该广义非线性算子的构造保证了高阶形变方程的线性特征．这一方法的有效性通过一些算例得到了验证．最后通过同伦分析方法对Butler-Volmer方程进行了求解,结果显示过电位和电流密度的级数解析解与数值解吻合很好,并有很好的收敛效率．     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

On Evolution Galerkin Methods for the Maxwell and the Linearized Euler Equations

The subject of the paper is the derivation and analysis of evolution Galerkin schemes for the two dimensional Maxwell and linearized Euler equations. The aim is to construct a method which takes into account better the infinitely many directions of propagation of waves. To do this the initial function is evolved using the characteristic cone and then projected onto a finite element space. We derive the divergence-free property and estimate the dispersion relation as well. We present some numerical experiments for both the Maxwell and the linearized Euler equations.     

Characteristic decompositions and boundary value problems for two‐dimensional steady relativistic Euler equations

In this paper, we investigate Goursat problems, and mixed initial and boundary value problems for the two‐dimensional steady relativistic Euler equations. The global existence of classical solutions to these problems are obtained by using the characteristic decomposition method. Some applications of these results in supersonic flow in two‐dimensional ducts and the two‐dimensional relativistic jet are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

The Inviscid Limit for the Steady Incompressible Navier-Stokes Equations in the Three Dimension*

In this paper, the authors consider the zero-viscosity limit of the three dimensional incompressible steady Navier-Stokes equations in a half space R⁺×R². The result shows that the solution of three dimensional incompressible steady Navier-Stokes equations converges to the solution of three dimensional incompressible steady Euler equations in Sobolev space as the viscosity coefficient going to zero. The method is based on a new weighted energy estimates and Nash-Moser itera...     

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.     

On positivity preserving finite volume schemes for Euler equations

Summary. We consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional Euler equations of gas dynamics. A general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate Riemann solver and the reconstruction are both positivity preserving. Appropriate limitation to achieve a high order positivity preserving reconstruction is described. Received May 20, 1994     

ON THE VISCOSITY SPLITT NG METHOD FOR INITIAL BOUNDARY VALUE PROBLEMS OF THE NAVIAR-STOKES EQUATIONS

The viscosily splitting method for the Navier-Stokes equations on two dimensional multi-connected domains is considered. The equation is split into an Euler equation and a non-stationary Stekes equation within each time step. The author proves the convergence theorem as he has done for the problem on simply connected domains, and the rate of convergence is improved from loss than 1/4 to 1.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.