三维Helmholtz方程的高阶隐式紧致差分方法

本文基于二阶导数的四阶Pade型紧致差分逼近式,并结合原方程本身,得到了三维Helmholtz方程的一种四阶精度的隐式紧致差分格式,该格式在每个空间方向上只涉及到三个点处的未知量及其二阶导数值。边界处对于二阶导数的离散格式利用四阶显式偏心格式。然后,利用Richardson外推法、算子插值法及二阶导数在边界点处的六阶显式偏心格式,将本文构造的格式精度提高到六阶。最后,通过数值实验验证了本文方法的精确性和可靠性。     

有阻尼体系动力分析的一种显式差分法

本文利用拉格朗日型的二次插值函数近似位移反应，建立了速度、加速度的差商近似公式，进而推得集中质量阻尼体系动力分析的一种显式差分法。该显式差分格式具有二阶计算精度，与目前常用的有阻尼体系动力求解的几种格式相比，具有明显减少计算工作量的优点。     

一类弱非线性波浪数值模型及其适用性分析

研究了一类含弱非线性的改进型Boussinesq水波方程,在非交错网格下,利用有限差分法建立了混合四阶Adams-Bashforth-Moulton的预报校正格式的波浪数值模型。在数值模型中,关于空间一阶导数差分格式采用四阶精度、二阶导数差分格式采用二阶精度。针对波浪的一维、二维传播变形问题进行了数值计算,并通过与相关实验结果对比分析考察了该数值模型的适用性。     

对称正则长波方程的拟紧致守恒差分格式

本文就对称正则长波方程的初边值问题进行了数值研究,提出了一个三层线性拟紧致差分格式,该格式具有较高精度且合理模拟了初边值问题的守恒性质。文章在先验估计基础上运用能量分析方法分析了格式的稳定性及二阶收敛性。数值结果验证了格式的有效性。     

二维对流扩散方程的二阶精度特征差分格式

针对二维对流扩散方程提出了几类二阶精度特征差分格式,给出了这些格式形成的线性代数方程组可解的充分条件,分析证明了这些格式按离散L^2模是二阶收敛的。最后,具体算例表明这些格式对于对流扩散方程有良好的计算效果。     

解KDV方程的一个二阶三层差分格式

本文讨论KDV方程定界问题的数值解法。对KDV方程构造了一个二阶三层的差分格式,并对非线性项进行了线性化,使格式的近似解更精确。通过严格的误差估计证明了非线性稳定性。数值实验结果表明了理论证明的正确性和格式的有效性。该格式是可行的,有推广价值。     

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。     

分解炉内三维湍流流场数值模拟

针对某大型高性能分解炉的实际尺寸，用Realizable(带旋流修正的)k-ε模型，采用SIMPLE算法，二阶迎风差分格式，模拟了炉内三维湍流流场．计算所得的速度、压力分布与该炉热工标定的结果以及其实际运行的性能符合较好．     

非等距网格上一维对流扩散方程的保正格式

对流扩散方程广泛存在于很多领域,为适应一些实际问题模型的求解,对离散格式,不仅要求满足一些基本性质,如稳定性和解的存在唯一性等,还要求离散格式的保正性．采用有限体积格式求解对流扩散方程的工作较少,但在保正性方面所做的工作不多．本文构造了任意非等距网格上一维对流扩散方程的非线性保正有限体积格式．其中,扩散通量的离散,在等距网格上,当扩散系数为标量时可退化为标准的二阶中心差分格式．而对流通量的离散,为避免数值振荡而使其保持迎风特性,提出一种新的方法使格式精度提高到二阶．该方法在上游单元中心处作泰勒级数展开,通过相关辅助未知量来完成梯度的重构,并对出负情形作正性校正,使得格式满足保正性要求．新格式只含有区间单元中心未知量,并满足区间端点处通量的局部守恒性．数值结果表明,本文所提格式是有效的,对于处理扩散占优、对流占优问题,扩散系数连续和间断情形均具有良好的适应性,并且保持二阶精度．另外,新格式适用于扩散系数间断问题的求解．     

抛物方程的一类区域分解迎风差分算法

本文给出了一阶迎风差分、内边界二阶迎风差分和二阶迎风差分三种格式的算法和误差估计。为了减弱稳定性条件限制,先在内边界点上采用小时间步长和大的空间步长进行多层计算,再在内点用隐格式并行计算。这些算法结合了迎风和区域分解的优点,计算格式简单,易于编程实现。     

Hybrid time domain solvers for the Maxwell equations in 2D

We present a new approach to time domain hybrid schemes for the Maxwell equations. By combining the classical FD‐TD scheme with two unstructured solvers, one explicit finite volume solver and one implicit finite element solver, we achieve a very efficient and flexible second‐order scheme. The second‐order accuracy of the hybrid scheme is verified through convergence studies on perfectly conducting as well as dielectric and diamagnetic circular cylinders. The numerical results also show its superiority to the FD‐TD scheme. Copyright © 2002 John Wiley & Sons, Ltd.     

A finite volume nodal point scheme for solving two dimensional Navier-Stokes equations

Summary A finite volume nodal point spatial discretization scheme for the computation of viscous fluxes in two dimensional Navier-Stokes equations has been presented here. The present scheme gives second order accurate first derivatives and at least first order accurate second derivatives even for stretched and skewed grid. It takes almost the same numerical efforts to solve full Navier-Stokes equations as that for using thin layer approximation. The scheme has been implemented to solve laminar viscous flows past NACA0012 aerofoil. To advance the solution in time a five stage Runge-Kutta scheme has been used. To accelerate the rate of convergence to steady state, local time stepping, residual averaging and enthalpy damping have been employed. Only a fourth order artificial dissipation has been used here for global stability of the solution. A comparative study of the results obtained by the present scheme for full Navier-Stokes equations and for thin layer approximation have been made with other numerical methods developed earlier.     

Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps

By introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.     

脉管制冷机工作过程数值模拟的Lax-Wendroff算法研究

提出将Lax—Wendroff数值方法应用于脉管制冷机气体流动和工作过程的数值模拟。分析了属于Lax—Wendroff数值方法类型、具有2阶精度格式、2阶段MacCormack算法及其相应的离散格式和算法组织。所进行的研究表明，将Lax-wen-droff算法类型应用于数值计算脉管制冷机工作过程所伴随的强烈热交换和流动损失现象的气体流动，可以获得很好的数值稳定性。     

Source prebiasing for improved second harmonic bubble-response imaging

The use of the second harmonic bandwidth in order to improve the contrast enhancement of vascular space provided by microbubble echo contrast is well established. A significant obstacle to improving on the contrast advantage of the second harmonic bandwidth arises from the linear response of tissue to the finite amplitude distortion produced second harmonic in the beam. A scheme in which the source wave contains a second harmonic component designed to cancel out the second harmonic produced by finite amplitude distortion in the focal region was computationally investigated. This prebiasing scheme was found to offer significant reductions in the amplitude of the second harmonic in the focal region. These reductions were found in both the homogeneous tissue path case and in the inhomogeneous tissue path case. The resulting clinical potential of source prebiasing is discussed. Also, it was observed that the inhomogeneous focusing of the finite amplitude distortion-produced second harmonic was significantly better than that of a same frequency fundamental with an identical homogeneous path focal profile.     

一类双曲型积分微分问题有限元逼近的超收敛估计

本文研究双曲型积分微分方程的半离散有限元逼近格式的超收敛估计.基于一种新的初值近似,得到了有限元解与精确解的Ritz-Volterra投影的Ws,p(Ω)模的如下超收敛估计k＞1,s=0,2≤p≤∞时,超收敛1阶;k＞1,s=1,2≤p＜∞时,超收敛2阶;k＞1,s=1,p=∞时,几乎超收敛2阶;k=1,s=1,2≤p ≤∞时,超收敛1阶.     

A finite difference scheme for solving a three‐dimensional heat transport equation in a thin film with microscale thickness

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a finite difference scheme with two levels in time for the three‐dimensional heat transport equation. It is shown by the discrete energy method that the scheme is unconditionally stable. The three‐dimensional implicit scheme is then solved by using a preconditioned Richardson iteration, so that only a tridiagonal linear system is solved each iteration. Numerical results show that the solution is accurate. Copyright © 2001 John Wiley & Sons, Ltd.