求解双曲型守恒律的半离散中心差分格式

给出了一种求解双曲型守恒律的三阶半离散中心差分格式。该格式以一种推广的三阶重构为基础,同时考虑了波传播的局部速度。格式的构造方法是利用重构,先计算非一致交错网格上的均值,再将该网格均值投影回原来的非交错网格,得到新的全离散中心差分格式,该格式有半离散形式。本文半离散格式保持了中心差分格式简单的优点,即不需用R iemann解算器,避免了进行特征解耦。它具有守恒形式,数值通量满足相容性条件。数值试验结果表明该格式是高精度、高分辨率的。     

二维双曲型守恒律的一种五阶松弛格式

松弛格式是Jin和Xin提出的无振荡有限差分方法,其主要思想是将守恒律转化为松弛方程组进行求解.本文用逐维五阶WENO重构和显隐式Runge-Kutta方法对松弛方程组的空间和时间进行离散,得到了一种求解二维双曲型守恒律五阶松弛格式.所得格式保持了松弛格式简单的优点,不用求解Riemann问题和计算通量函数的雅可比矩阵.通过二维Burgers方程和二维浅水方程的数值算例验证了格式的有效性.     

求解多维双曲守恒律方程组的四阶半离散格式

提出了求解多维双曲守恒律方程组的四阶半离散格式。该方法以中心加权基本无振荡(CWENO)重构为基础,同时考虑到在R iemann扇内波传播的局部速度,从而回避了计算过程中的网格交错,建立了数值耗散较小的介于迎风格式和中心格式之间的半离散格式。本文的四阶半离散格式是Kurganov等人的三阶半离散格式的高阶推广。大量的数值算例充分说明了本文方法的高分辨率和稳定性。     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

一般坐标系下的二维Euler方程时—空守恒格式

将文「１－３」中的一维时－空守恒格式推广到了二维情形,得到了一般坐标系下的二维Ｅｕｌｅｒ方程时－空守恒格式,并用几个典型算例进行了检验计算,结果表明：本文得到的二维时－空守恒格式保留了一维格式所有的优点,格式简单,通用性强,而且对激波等间断具有很高的分辨率。     

基于Level Set的多维双曲守恒律标量方程的激波追踪方法

结合四阶CWENO(Cemral Weighted Essentially Non-Oscillatory)格式、四阶NCE(Natural Continuous Extensions)Runge-Kutta法和Level Set方法，很好地处理了一维双曲守恒律标量方程的激波追踪问题。针对二维双曲守恒律标量方程，成功地用五阶WENO格式、非TVD格式的四阶Runge-Kutta方法和Level Set方法进行激波追踪。将所得的数值解与标准的高阶激波捕捉方法所得的数值解进行比较，说明基于Level Set的激波追踪方法的有效性与逐点收敛性。     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

一般形式的二维时—空守恒格式

本文将经作者改进后的一维时－空守恒格式推广到了二维情形，得到了一个一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式，该格式对各种不规则几何区域内的流动问题具有很强的适应性，同时它还保留了一维格式的优点。几个典型算例的计算结果表明，本文格式不仅精度高，通用性好，而且对激波等间断具有很高的分辨率。     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

二维时-空守恒格式及其在激波传播计算中的应用

将改进后的一维时－空守恒格式推广到了二维情形,得到了一个新的一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式,并用该格式对几个具有复杂波系的流场进行了数值模拟。结果表明,该格式保留了一维格式通用性好、结构简单的优点,其计算结果精度高,对激波等间断具有很强的分辨率。     

Third-order modified coefficient scheme based on essentially non-oscillatory scheme

A third-order numerical scheme is presented to give approximate solutions to multi-dimensional hyperbolic conservation laws only using modified coefficients of an essentially non-oscillatory （MCENO） scheme without increasing the base points during construction of the scheme. The construction process shows that the modified coefficient approach preserves favourable properties inherent in the original essentially nonoscillatory （ENO） scheme for its essential non-oscillation, total variation bounded （TVB）, etc. The new scheme improves accuracy by one order compared to the original one. The proposed MCENO scheme is applied to simulate two-dimensional Rayleigh-Taylor （RT） instability with densities 1：3 and 1：100, and solve the Lax shock-wave tube numerically. The ratio of CPU time used to implement MCENO, the .third-order ENO and fifth-order weighed ENO （WENO） schemes is 0.62：1：2.19. This indicates that MCENO improves accuracy in smooth regions and has higher accuracy and better efficiency compared to the original ENO scheme.     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases.     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases. The project supported by the China NKBRSF (2001CB409604) The English text was polished by Yunming Chen     

Characteristic-based finite volume scheme for 1D Euler eouations

In this paper, a high-order finite-volume scheme is presented for the one-dimensional scalar and inviscid Euler conservation laws. The Simpson's quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson's quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory (CWENO) reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.     

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.     

Harten-Lax-van Leer-contact (HLLC) approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver is built with elastic waves (HLLCE) for one-dimensional elastic-plastic flows with a hypoelastic constitutive model and the von Mises’ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the presented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves (TRRSE).     

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the one- dimensional scalar and inviscid Euler conservation laws. The Simpson＇s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson＇s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory （CWENO） reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.     

叶栅可压缩流场的数值分析

将一维ＡＵＳＭ＾＋格式与三阶精度ＭＵＳＣＬ格式融合,给出了其在任意风线坐标下的二维形式,采用有限体积法,将ＡＵＳＭ＾＋格式与Ｒｕｎｇｅ－Ｋｕｔｔａ格式结合,对叶栅可压缩流场进行了数值模拟,典型算例的计算结果与有关献结果相符很好,表明ＲＫ－ＡＵＳＭ＾＋混合格式对叶栅可压缩流场进行数值模拟是可行的。     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.