摄动有限差分方法研究进展

振动有限差分（ＰＦＤ）方法,既离散徽商项也离散非微商项（包括微商系数）,在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式,它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．     

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional(2D) tank are studied.The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid.A time-independent finite difference method,which is developed by Bang-fuh Chen,is used to solve the Euler equations for incompressible and inviscid fluids.The numer...     

一个高分辨率的矢通量分裂—TVD杂交新格式

本文提出了一个新的杂交格式,它将Steger-Warming的矢通量分裂与Harten的TVD格式紧密结合在一起,构造了一个高分辨率的新格式,用于计算跨声速流场和捕获激波。典型的定常跨声速叶栅流算例表明:当Courant数取6～100时,一般在90步内残差的二范数下降三个数量级,这样的收敛率要比Beam-Warming格式快得多;观察残差的收敛历史发现:收敛曲线并无大的波动;分析激波附近的数值结果,没出现“低亏,过跳”、伪振荡现象,叶盆和叶背面上激波前或激波后的参数无波动;在60×15网格下捕获的激波过渡区不超过2个网格,表明了该格式具有较高的分辨率,能在不人为附加耗散项的条件下给出高质量无数值波动的激波流场解。     

New numerical schemes based on a criterion for constructing essentially stable and accurate numerical schemes for convection-dominated equations

In order to obtain stable and accurate numerical solutions for the convection-dominated steady transport equations, we propose a criterion for constructing numerical schemes for the convection term that the roots of the characteristic equation of the resulting difference equation have poles. By imposing this criterion on the difference coefficients of the convection term, we construct two numerical schemes for the convection-dominated equations. One is based on polynomial differencing and the other on locally exact differencing. The former scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $, which is the critical value for its stability, while it approaches the second-order upwind scheme as Rm goes to infinity. Hence the former scheme interpolates a stable scheme between the QUICK scheme at Rm = $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $ and the second-order upwind scheme at Rm = ∞. Numerical solutions with the present new schemes for the one-dimensional, linear, steady convection-diffusion equations showed good results.     

Retrospective time integral scheme and its applications to the advection equation

To put more information into a difference scheme of a differential equation for making an accurate prediction, a new kind of time integration scheme, known as the retrospective (RT) scheme, is proposed on the basis of the memorial dynamics. Stability criteria of the scheme for an advection equation in certain conditions are derived mathematically. The computations for the advection equation have been conducted with its RT scheme. It is shown that the accuracy of the scheme is much higher than that of the leapfrog (LF) difference scheme. The project supported by the National Key Program for Developing Basic Sciences (G1999043408 and G1998040901-1) and the National Natural Sciences Foundation of China (40175024 and 40035010)     

小波插值方法自适应数值求解时间进化微分方程

应用小波自相关函数的插值性质,得到任意给定函数的插值小波表达式,然后对其直接求导,可以得到函数导数的表达式。导数运算不再应用差分算法,扩展了小波方法在数值求解微分方程中的应用。由于小波基函数的有限支撑特点,小波方法可以有效地处理微分方程中解的局部突变问题。通过设定小波系数阀值,实现了求解过程的自适应。本文给出了两个算例,结果表明了算法的自适应特点及其向二维空间问题推广的有效性。     

求解气动方程的混合反扩散方法

本文综述了反扩散格式的发展,指出利用混合反扩散方法,理论上既可导出Beam—Warming以及Jameson等发展的含参数的格式,又可导出不含参数的TVD格式.研究了含参数的混合反扩散格式和不含参数的反扩散格式,并介绍了格式的应用情况.