求解Navier-Stokes方程组的组合紧致迎风格式

给出一种新的至少有四阶精度的组合紧致迎风(CCU)格式,该格式有较高的逼近解率,利用该组合迎风格式,提出一种新的适合于在交错网格系统下求解Navier-Stokes方程组的高精度紧致差分投影算法.用组合紧致迎风格式离散对流项,粘性项、压力梯度项以及压力Poisson方程均采用四阶对称型紧致差分格式逼近,算法的整体精度不低于四阶.通过对Taylor涡列、对流占优扩散问题和双周期双剪切层流动问题的计算表明,该算法适合于对复杂流体流动问题的数值模拟.     

求解可压Navier-Stokes方程的对称超紧致差分格式及其并行算法

考查了超紧致差分方法,并将其精度同传统差分格式和紧致差分格式做了比较,结果显示超紧致方法具有良好求解效率.用分块流水线方法设计了超紧致差分格式的并行算法,进行数值实验及并行性能分析.     

求解对流扩散方程的紧致修正方法

提出了求解对流扩散方程的紧致修正方法,该方法是在低阶离散格式的源项中,引入紧致修正项,从而构造高阶紧致修正格式,并进行求解.采用紧致修正方法对典型的对流扩散方程进行计算.结果表明,紧致修正方法虽然与二阶经典差分方法建立在相同的结点数上,但紧致修正方法的精度与紧致方法的精度相同,均具有四阶精度.所以紧致修正方法可以在少网...     

四阶紧致差分格式对静力模式中垂直网格计算频散性的影响

为了说明四阶紧致差分格式在大气和海洋数值模式中的潜在价值,提出一种通用方法,推导静力线性斜压适应方程组在微分和差分情况下的频散关系,水平尺度分100 km,10 km和1 km三种情况,从频率、水平群速和垂直群速方面,对采用二阶中央差和四阶紧致差分格式情况下,非跳点网格(N网格)、Lorenz网格(L网格)、Charney-Phillips网格(CP网格)、Lorenz时间跳点网格(LTS网格)和Charney-Phillips时间跳点网格(CPTS网格)的计算特性进行比较,发现采用高精度的四阶紧致差分格式总体上可以明显减少上述三种水平尺度波动在N网格、CP网格、L网格和CPTS网格上的频率、水平群速和垂直群速误差,但对LTS网格,采用四阶紧致差分格式,会使得计算水平群速和垂直群速误差变大.     

气动计算中色散可控的迎风紧致格式

文中通过对修正方程色散项的耗散类比方法,指出该项在改善数值解中非物理振荡的重要作用,给出了一类依赖于三个自由参量的色散可控迎风紧致格式。通过这三个参量可控制耗散量的大小,也可控制色散量的大小及方向,并给出了一个具体的色散协调因子。文中给出的格式有着精度高、方法简单、计算量小和有着强的对激波的捕捉能力等优点。对二维激波反射问题进行了数值实验。计算结果非常令人满意。     

一类迎风紧致差分格式全离散特性分析

采用Ｆｏｕｒｉｅｒ分析方法，通过显式多步Ｒｕｎｇｅ－Ｋｕｔｔａ时间离攻一维线性对流方程，导出了一类高精度迎风紧致格式全离散色散关系式，详细分析了不同ＣＦＬ数下所研究差分格式的耗散、色散及相应的相速度、群速度等特性。以数值实验显示了格式较高的计算精度和分辨率。     

随机扰动对拟小波方法求解对流扩散方程的影响

引进拟小波方法数值求解对流扩散方程，研究结果表明，计算带宽W有一个极值，当计算带宽W取该极值时，该方程的拟小波解的精度最高，且好于迎风格式。当边界发生随机不等幅扰动时，对于积分时间较长的情况，拟小波格式的效果要稍逊于迎风格式；当边界发生随机等幅扰动时，若计算带宽W取大于等于20的整数时，方程拟小波解的精度与迎风格式相同；当参数受到随机扰动时，W取10时的拟小波解的均方根误差要小于迎风格式；在初值发生随机扰动且计算带宽W取10时，方程的拟小波解的精度最高，好于迎风格式。     

耗散紧致格式的频谱特性研究与应用

采用Fourier分析方法,给出线性耗散紧致格式和基于m级Runge-Kutta时间积分方法的全离散格式的频谱特性,并应用五阶耗散紧致格式模拟典型高频波传播和超声速平面Couette流动的特征值问题及其稳定性边值问题,展示耗散紧致差分格式良好的频谱特性.     

基于气动声学问题的高阶非线性紧致格式

文章基于线性中心紧致差分格式, 通过非线性加权插值的方法来求解网格中心处的函数值.这类格式保持了原有中心紧致差分格式的高阶精度和低耗散特性, 同时其分辨率也非常高, 由于其非线性插值的机制, 使得这类格式能够捕捉强激波, 所以这类新的高阶非线性紧致格式是一种较好的模拟湍流和气动声学等多尺度问题的方法.      

基于SIMPLE的紧致方法

通过泰勒展式系数匹配的方法发展了基于非等距网格的有限容积紧致格式,采用延迟修正的方法建立了基于SIMPLE的紧致方法,,该方法能够得到高精度的数值解,增加迭代求解代数方程组的稳定性。对底部加热的方腔内自然对流换热问题进行数值模拟,结果表明,紧致方法比二阶中心差分方法具有更高的精度。     

A new family of high-order compact upwind difference schemes with good spectral resolution

This paper presents a new family of high-order compact upwind difference schemes. Unknowns included in the proposed schemes are not only the values of the function but also those of its first and higher derivatives. Derivative terms in the schemes appear only on the upwind side of the stencil. One can calculate all the first derivatives exactly as one solves explicit schemes when the boundary conditions of the problem are non-periodic. When the proposed schemes are applied to periodic problems, only periodic bi-diagonal matrix inversions or periodic block-bi-diagonal matrix inversions are required. Resolution optimization is used to enhance the spectral representation of the first derivative, and this produces a scheme with the highest spectral accuracy among all known compact schemes. For non-periodic boundary conditions, boundary schemes constructed in virtue of the assistant scheme make the schemes not only possess stability for any selective length scale on every point in the computational domain but also satisfy the principle of optimal resolution. Also, an improved shock-capturing method is developed. Finally, both the effectiveness of the new hybrid method and the accuracy of the proposed schemes are verified by executing four benchmark test cases.     

A new family of high-order compact upwind difference schemes with good spectral resolution

This paper presents a new family of high-order compact upwind difference schemes. Unknowns included in the proposed schemes are not only the values of the function but also those of its first and higher derivatives. Derivative terms in the schemes appear only on the upwind side of the stencil. One can calculate all the first derivatives exactly as one solves explicit schemes when the boundary conditions of the problem are non-periodic. When the proposed schemes are applied to periodic problems, only periodic bi-diagonal matrix inversions or periodic block-bi-diagonal matrix inversions are required. Resolution optimization is used to enhance the spectral representation of the first derivative, and this produces a scheme with the highest spectral accuracy among all known compact schemes. For non-periodic boundary conditions, boundary schemes constructed in virtue of the assistant scheme make the schemes not only possess stability for any selective length scale on every point in the computational domain but also satisfy the principle of optimal resolution. Also, an improved shock-capturing method is developed. Finally, both the effectiveness of the new hybrid method and the accuracy of the proposed schemes are verified by executing four benchmark test cases.     

A 5th order monotonicity-preserving upwind compact difference scheme

Based on an upwind compact difference scheme and the idea of monotonicity-preserving, a 5th order monotonicity-preserving upwind compact difference scheme (m-UCD5) is proposed. The new difference scheme not only retains the advantage of good resolution of high wave number but also avoids the Gibbs phenomenon of the original upwind compact difference scheme. Compared with the classical 5th order WENO difference scheme, the new difference scheme is simpler and small in diffusion and computation load. By emplo...     

Higher Order KFVS Algorithms Using Compact Upwind Difference Operators

A family of high order accurate compact upwind difference operators have been used, together with the split fluxes of the KFVS (kinetic flux vector splitting) scheme to obtain high order semidiscretizations of the 2D Euler equations of inviscid gas dynamics in general coordinates. A TVD multistage Runge–Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic flow problems which indicate the higher accuracy and low diffusion realizable in such schemes.     

三维边界层内定常横流涡的感受性研究

层流向湍流转捩的预测与控制一直是研究的前沿热点问题之一,其中感受性阶段是转捩过程中的初始阶段,它决定着湍流产生或形成的物理过程.但是有关三维边界层内感受性问题的数值和理论研究都比较少;实际工程问题中大部分转捩过程都是发生在三维边界层流中,所以研究三维边界层中的感受性问题显得尤为重要.本文以典型的后掠角45?无限长平板为例,数值研究了在三维壁面局部粗糙作用下的三维边界层感受性问题,探讨了三维边界层感受性问题与三维壁面局部粗糙长、宽和高之间的关系;然后,考虑在后掠平板上设计不同的三维壁面局部粗糙的分布状态、几何形状、距离后掠平板前缘的位置以及流向和展向设计多个三维壁面局部粗糙对三维边界层感受性问题有何影响;最后,讨论两两三维壁面局部粗糙中心点之间的距离以及后掠角的改变对三维边界层感受性的物理过程将会发生何种影响等.这一问题的深入研究将为三维边界层流中层流向湍流转捩过程的认识和理解提供理论依据.     

迎风紧致格式求解Hamilton-Jacobi方程

基于Hamilton-Jacobi(H-J)方程和双曲型守恒律之间的关系,将三阶和五阶迎风紧致格式推广应用于求解H-J方程,建立了高精度的H-J方程求解方法.给出了一维和二维典型数值算例的计算结果,其中包括一个平面激波作用下的Richtmyer Meshkov界面不稳定性问题.数值试验表明,在解的光滑区域该方法具有高精度,而在导数不连续的不光滑区域也获得了比较好的分辨效果.相比于同阶精度的WENO格式,本方法具有更小的数值耗散,从而有利于多尺度复杂流动的模拟中H-J方程的求解.     

Further improvement and analysis of CCD scheme: Dissipation discretization and de-aliasing properties

In this paper, we further analyze a combined compact difference (CCD) scheme proposed recently [T.K. Sengupta, V. Lakshmanan, V.V.S.N. Vijay, A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys. 228 (8) (2009) 3048–3071] for its dissipation discretization properties to show that its superiority also helps in controlling aliasing error for a benchmark internal flow. However, application of the same CCD method to study the receptivity of a boundary layer experiencing adverse pressure gradient is not successful. This is traced to the nature of the equilibrium flow where the better dissipation property is not helpful in the inviscid part of the flow, where the aliasing problems continue to persist. A further modification is proposed to the CCD method here to solve complex physical problems requiring information on higher order disturbance quantities – as in problems of flow receptivity and instability.     

非结构网格上的迎风有限元格式

在非结构网格上提出一种基于修正积分区域的迎风有限元格式,它与一阶迎风差分格式相当,可应用于构造各种不同的数值格式。