流体力学高精度高分辨率差分格式的研究
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摘要
在计算流体力学中，对同时包含间断和复杂流动现象的流场进行数值模拟是一个迫切的并且也是非常困难的任务。近年来，针对此类问题的捕捉激波的高精度高分辨率计算方法得到了迅速的发展。而在流体力学数值算法中，有限差分方法历史最悠久也最为成熟，并且适宜于构造高精度的格式。本论文主要研究流体力学高精度高分辨率差分格式，对格式的精度和稳定性进行了分析和讨论。论文内容主要包含以下几个方面：1. WENO格式的精度以及光滑因子的研究
给出了一个方便使用的五点WENO格式达到最高精度的充分条件，以及一般情况下的WENO格式的精度公式。另外，对Jiang和Shu的WENO格式的光滑因子进行了分析，给出了两个低耗散的光滑因子，并分析了Liu等人的WENO格式的光滑因子，通过Borges等人的WENO权因子计算方法保证收敛精度。数值算例表明，使用这些光滑因子的WENO格式对连续流场的捕捉能力有所提高。2.紧致格式及紧致WENO混合格式的研究
构造了一个使用对称网格模板的五阶守恒型迎风紧致格式，与经典的五阶迎风紧致格式相比，该格式耗散较低，且拥有更大的稳定性范围，并且给出了一种显式的保证整体精度为五阶的边界格式。另外，对Ren等人的紧致WENO混合格式做出了改进，对欧拉方程采用了新的处理，从而避免了块三对角方程组的求解，大大降低了计算量。3.简化了半离散差分格式稳定性条件
傅立叶分析方法得到的稳定性条件一般为余弦函数的不等式，不易于使用。本文将余弦函数转化为多项式形式，同时证明了当差分格式每增加两阶收敛精度，该多项式就可以提取一个因子，这使得该多项式不等式的求解大为简化。该方法对判断半离散差分格式的稳定性十分有效。而在构造高精度优化格式时，该方法得到的稳定性条件可以表示为差分格式系数的有限个约束的形式，这使得在优化过程中加入稳定性限制变得可行。4.优化型格式的研究
通过将一个给定模板上的最高阶精度格式与一种优化格式进行加权平均，权系数由本文提出的局部波数指示子确定，得到了一种保精度优化（Maximum order preserving optimized,MOPO）格式。然后将这种格式与六点对称WENO格式融合，得到了一种MOPOWENO格式。数值结果表明，MOPO格式与MOPOWENO格式不仅保持了最高阶精度，并且在所测试问题中比相应的两种子格式的相位误差更小。
In computational fluid dynamics, the numerical simulation of flow with both discontinuities andcomplex structures is an urge and difficult task. To solve these problems, high-order high-resolutionnumerical methods have been proposed recently. In the numerical methods of fluid dynamics, finitedifference schemes are simple and easy to achieve high order, therefore they have been extensivelystudied. The main object of this thesis is to find difference schemes which are more suitable for aboveproblems. Besides that, there are some discussions about the convergence order and the stability ofdifference schemes. The main content includes the following parts:1. Research about the convergence order and the smoothness indicator of WENO schemes
A sufficient condition for five-points WENO schemes achieving the fifth-order is given which iseasy to use. And the formula of accuracy orders of WENO schemes is also given. Furthermore, thesmoothness indicator of Jiang and Shu is improved for better resolution on complex structure. Inaddition to this, the first smoothness indicator proposed by Liu et al. is analyzed in detail.2. Research about compact schemes and hybrid compact WENO schemes
A fifth-order conservative upwind compact schemes and an explicit boundary scheme whichmaintains the global fifth-order accuracy are given. Furthermore, based on the fifth-order hybridcompact WENO scheme proposed by Ren et al., a new treatment for Euler equations is given. Sincethis treatment has avoided the block tridiagonal equations, it reduces the computation costs greatly. Atthe same time, it increases the dissipation slightly because of its global flux splitting.3. Research about stability criterion of semi-discrete difference schemes
The stability condition for semi-discrete difference schemes obtained from Fourier analysis isusually an inequality of trigonometric function which is not convenient to use. For difference schemeon uniform grids, this trigonometric function can be converted into a polynomial form. Taking theconvergence order of the scheme into consideration, this polynomial can be factorized into a simpleform. Thus, the corresponding inequality is much easier to solve than the original one. This method isvery effective for judging the stability of semi-discrete difference schemes. Particularly, by using thismethod, it is practicable to add the stability constraints when designing high-order schemes.4. Research about optimized scheme and optimized WENO schemes
A maximum order preserving optimized (MOPO) scheme which is an weighted average of themaximum order scheme and an optimized scheme is given. To determine the weights of these two linear weights, a local wavenumber indicator is introduced. Furthermore, a maximum order preservingoptimized WENO (MOPOWENO) scheme is achieved by integrating it with the sixth-points WENOschemes framework. Numerical results show that the proposed schemes combined both advantages ofthe maximum order schemes and the optimized schemes.
给出了一个方便使用的五点WENO格式达到最高精度的充分条件，以及一般情况下的WENO格式的精度公式。另外，对Jiang和Shu的WENO格式的光滑因子进行了分析，给出了两个低耗散的光滑因子，并分析了Liu等人的WENO格式的光滑因子，通过Borges等人的WENO权因子计算方法保证收敛精度。数值算例表明，使用这些光滑因子的WENO格式对连续流场的捕捉能力有所提高。2.紧致格式及紧致WENO混合格式的研究
构造了一个使用对称网格模板的五阶守恒型迎风紧致格式，与经典的五阶迎风紧致格式相比，该格式耗散较低，且拥有更大的稳定性范围，并且给出了一种显式的保证整体精度为五阶的边界格式。另外，对Ren等人的紧致WENO混合格式做出了改进，对欧拉方程采用了新的处理，从而避免了块三对角方程组的求解，大大降低了计算量。3.简化了半离散差分格式稳定性条件
傅立叶分析方法得到的稳定性条件一般为余弦函数的不等式，不易于使用。本文将余弦函数转化为多项式形式，同时证明了当差分格式每增加两阶收敛精度，该多项式就可以提取一个因子，这使得该多项式不等式的求解大为简化。该方法对判断半离散差分格式的稳定性十分有效。而在构造高精度优化格式时，该方法得到的稳定性条件可以表示为差分格式系数的有限个约束的形式，这使得在优化过程中加入稳定性限制变得可行。4.优化型格式的研究
通过将一个给定模板上的最高阶精度格式与一种优化格式进行加权平均，权系数由本文提出的局部波数指示子确定，得到了一种保精度优化（Maximum order preserving optimized,MOPO）格式。然后将这种格式与六点对称WENO格式融合，得到了一种MOPOWENO格式。数值结果表明，MOPO格式与MOPOWENO格式不仅保持了最高阶精度，并且在所测试问题中比相应的两种子格式的相位误差更小。
In computational fluid dynamics, the numerical simulation of flow with both discontinuities andcomplex structures is an urge and difficult task. To solve these problems, high-order high-resolutionnumerical methods have been proposed recently. In the numerical methods of fluid dynamics, finitedifference schemes are simple and easy to achieve high order, therefore they have been extensivelystudied. The main object of this thesis is to find difference schemes which are more suitable for aboveproblems. Besides that, there are some discussions about the convergence order and the stability ofdifference schemes. The main content includes the following parts:1. Research about the convergence order and the smoothness indicator of WENO schemes
A sufficient condition for five-points WENO schemes achieving the fifth-order is given which iseasy to use. And the formula of accuracy orders of WENO schemes is also given. Furthermore, thesmoothness indicator of Jiang and Shu is improved for better resolution on complex structure. Inaddition to this, the first smoothness indicator proposed by Liu et al. is analyzed in detail.2. Research about compact schemes and hybrid compact WENO schemes
A fifth-order conservative upwind compact schemes and an explicit boundary scheme whichmaintains the global fifth-order accuracy are given. Furthermore, based on the fifth-order hybridcompact WENO scheme proposed by Ren et al., a new treatment for Euler equations is given. Sincethis treatment has avoided the block tridiagonal equations, it reduces the computation costs greatly. Atthe same time, it increases the dissipation slightly because of its global flux splitting.3. Research about stability criterion of semi-discrete difference schemes
The stability condition for semi-discrete difference schemes obtained from Fourier analysis isusually an inequality of trigonometric function which is not convenient to use. For difference schemeon uniform grids, this trigonometric function can be converted into a polynomial form. Taking theconvergence order of the scheme into consideration, this polynomial can be factorized into a simpleform. Thus, the corresponding inequality is much easier to solve than the original one. This method isvery effective for judging the stability of semi-discrete difference schemes. Particularly, by using thismethod, it is practicable to add the stability constraints when designing high-order schemes.4. Research about optimized scheme and optimized WENO schemes
A maximum order preserving optimized (MOPO) scheme which is an weighted average of themaximum order scheme and an optimized scheme is given. To determine the weights of these two linear weights, a local wavenumber indicator is introduced. Furthermore, a maximum order preservingoptimized WENO (MOPOWENO) scheme is achieved by integrating it with the sixth-points WENOschemes framework. Numerical results show that the proposed schemes combined both advantages ofthe maximum order schemes and the optimized schemes.
引文
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