数值方程

Firstly,the differential equation with parameters is given,at the same time,in order to let the truncation error of differential equation reach O(Δt~2+Δx~4) as high as possible,the method of undetermined coefficent is used to appropriately select the expression of these parameters.Secondly,the stability condition of the scheme derived is given by using the Fourier method of stability analysi...

首先给出了含参变量的差分方程，并用待定系数法适当地选取了这些参数的表示式，以使差分方程的截断误差阶尽可能高地达到了O(Δt2+Δx4)；其次用稳定性分析的Fourier方法给出了所得格式的稳定性条件；然后给出了确定差分格式中参数的两种方法，得到了一个稳定性条件为r/3的分支稳定的高精度显式差分格式；最后给出了数值例子，数值结果表明了本文格式较现有同类格式的优越性和理论分析的正确性。

In this paper, we construct the differential equation to solve the unconstrained optimization problem. Such methods solving the ill-conditioned problems are very effective. It indicates that the asymptotically stable region of explicit the Euler method to solve the differential equation equals to the descent region of the gradient method to solve the original unconstrained optimization problem.

我们发现，在用Euler方法求解二次优化问题的等价动力系统的方程时，由方法的步长确定的稳定区域对应于这些方法所得到的迭代公式的步长满足单调下降算法的条件确定的单调下降区域，因此我们可以利用这个性质构造解无约束优化问题的数值方法而不采用标准的常微分方程的数值求解公式。

According to the properties of Wiener increments and the independent relation between the Wiener increments and the numerical solutions, the difference equation obtained by applying the numerical method to the general linear test equation is studied carefully in mean square sense and the conditions under which the numerical method is MS-stable and GMS-stable are determined.

依据维纳增量的性质以及维纳增量和数值解间的独立关系在均方意义下详细分析了数值方法应用到线性试验方程上得到的差分方程，并进一步确定了半隐式Milstein方法MS-稳定和GMS-稳定的条件。

The axially symmetric torsion of rigid circular shaft of varying diameter embedded in an elastic half space is studied by line-loaded integral equation method, where the problem is formulated by distributions of fictitions fundamental loads PRCHS (point ring couple in half space) along the axis of symmetry in interval of the shaft and is reduced to a one-dimensional and non-singular Fredholm integral equation of the first kind and is easily solved numerically.

弹性半空间；本文用线载荷积分方程法研究嵌在弹性半空间的刚性变直径圆轴的轴对称扭转。利用将"半空间的点环力偶"这一虚的基本载荷沿对称轴的圆轴区间中分布就能使本问题归结为一个一维的非奇异的Ferdholm第一种积分方程，且很易用数值求解。文中给出刚性圆锥，圆柱，圆锥柱嵌在弹性半空间的扭转的数值计算的例并和他人的已知结果相比较。

By the theory of natural boundary reduction, the natural integral equation and the Poisson integral formula of the problem considered are obtained, and the numerical method of the natural integral equation is given.

通过自然边界归化，获得了该问题的自然积分方程和Poisson积分公式，给出了自然积分方程的数值解法，并通过数值例子以示本文方法的可行性与有效性。

Firstly, Wavelet-Galerkin algorithm for solving the first kind of singular integral equation with the Hilbert kernel is proposed, we use the characteristic of periodic wavelet on L~2([0,1]) and Hilbert kernel to solve and make stiff matrix lower dimensions and become sparser through thresholding,thus the cost of computation is reduced. Because of the singularity of Hilbert kernel we use Tikhonov regularization method to solve the system of stiff equation. At last the convergence and numerical result of approximate solution are given. Secondly, an approach of regularization based on Fourier is presented for sideways heat equation; we give the theory proof and error estimate.

首先，提出了含Hilbert核的第一类奇异积分方程的小波伽辽金(Wavelet-Galerkin)数值算法，该算法中利用了L~2([0，1])上的周期小波和Hilbert核的特点进行处理，使得刚性矩阵维数降低并且通过阈值使得它更加稀疏，减少了计算量；由于Hilbert核的奇异性，通过Tikhonov正则化方法求解所得到的刚性方程组，给出了收敛性和数值结果；其次，对标准的一维逆热传导方程给出了一种基于Fourier正则化方法，给出了理论证明及其误差估计，解决了文献中算法与理论误差估计的不相匹配的现象，该正则化方法不仅保留了测量数据的部分高频成份，且与文献中的算法具有同样的计算量和误差估计。

In the low-low satellite to satellite gravity mission, the intersatellite range and range-rate data are the observable values, while the intersatellite acceleration can be derived from the range-rate through digital differential. Using acceleration data, people can avoid solving the variation equations of the satellite motion and simplify the observation equations. But on the other side, the digital differential will amplify the observation noises and finally affect the solved geopotential accuracy.

低低卫星跟踪卫星的观测量是两低轨卫星的星间距离或星间速度，星间加速度由星间速度通过数值微分导出，用星间加速度作为观测量可以避免解算卫星运动的变分方程，简化观测方程的建立，但数值微分会使观测噪声放大，从而影响重力位的解算精度。

By adapting the existed numerical methods for ordinary and delay differential equations, and matching certain suitable numerical quadrature formulas, Rosenbrock methods are constructed for delay integro-differential equations, the stability criteria of the methods are derived.

通过改造现有常及离散型延迟微分方程的数值方法，并匹配以适当数值求积公式，构造了求解时滞积分微分方程的Rosenbrock方法，导出了其稳定性准则。

Based on this, a new WENO difference scheme which based on Dispersion-Relation-Preserving relation is developed, and representative test cases with this scheme for computational aeroacoustics problems has been implemented and compared in order to test capability of wave capturing; In addition, WENO schemes generally do not converge at high order in the presence of contact discontinuity of Euler equations, so a conservative front tracking technique coupling WENO schemes and Level Set method to simulate the translating density profile is presented here, and numerical simulation with this technique for representative test case has been implemented and results show the desired accuracy.

本文研究了高阶精度加权基本无振荡格式及其在双曲守恒律方程中的应用，在此基础上作了两个方面的工作：一是针对高频声波问题构造出一种基于保色散关系的WENO有限差分格式，并对计算气动声学问题的代表性算例进行了大量数值实验，比较了该格式捕捉波数的能力；另外，针对高阶WENO格式在处理Euler方程的接触间断时精度有所降低的问题，研究了利用界面追踪技术Level Set方法和高阶激波捕捉WENO格式相结合的一种守恒追踪方法，并且给出有代表性的密度滑移面问题的算例，得到一致高阶精度的数值模拟结果。

However, the multi-time values before the initial time are required in solving the ordinary differential equation with linear multi-step method, the values at these time levels are still computed from the given initial value by one of above mentioned schemes.

在常微分方程的线性多步法中用到初始时次前若干时次的数值，但这些时次上的起动值仍然由初值运用方程计算得到的。这就是说，人们在预报问题中始终恪守只用一个初值的原则。在天气预报中，数值积分只用到一个起报场，即最后一个时次的观测场。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。