IMECH-IR
欧拉方程的高效高精度时间及空间算法研究
Alternative TitleThe Study of High Efficiency and High Accuracy Numerical Methods for Euler Equations
唐荣
Thesis Advisor崔凯
2019-05-26
Degree Grantor中国科学院大学
Place of Conferral北京
Subtype硕士
Degree Discipline流体力学
Keyword欧拉方程 共享权重 Weno 格式 多步加权 Crweno 格式 最后一级隐式 Runge-kutta 方法
Abstract

复杂流动的大规模、长时间数值模拟,除了要求一定的精度和分辨率之外,还要求
其计算过程具有较强的稳定性及较高的计算效率,即需要发展高效高精度的时间及空间
算法。加权基本无振荡(Weighted essentially non-oscillatory,简称 WENO)格式因其具
有优秀的高精度激波捕捉能力而成为当下应用和发展最为广泛的流动空间离散格式;而
多级 Runge-Kutta 方法具有精度高、稳定性好、存储量低等特点,作为经典的常微分方
程数值解法,常被用于计算流体力学领域的时间方向离散推进计算。本文以离散 WENO
空间格式以及 Runge-Kutta 时间推进方法为基础,发展了计算效率更高,而且数值耗散
误差小、稳定性强的相应方法。主要工作包括:
1)发展了一种针对求解流动 Euler 方程的共享权重 WENO 方法,该方法采用
Steger-Warming 分裂后的能量通量来计算 WENO 格式中的子模板权值,所有通量分量
皆利用该权值进行加权组合,因而极大地提高了 Euler 方程的计算效率。对于共享权重
WENO 方法,加权变量的选取起着十分重要的作用,不当的变量选取会导致格式的稳
定性不足或分辨率降低。数值实验结果表明本文所发展的格式具有较好的鲁棒性以及较
低的耗散误差。
2)进一步发展了多步加权 CRWENO 格式。多步 CRWENO 格式在光滑区域具有紧
致格式的谱分辨性质,同时在间断点附近具有较高的精度和分辨率,但与多步 WENO
格式类似,最初的多步 CRWENO 格式同样存在计算量大的问题。本文将最近曾方军等
人提出的关于改进多步 WENO 格式的思想和方法推广到多步 CRWENO 格式,在保证
精度不降低的情况下极大地提高了多步 CRWENO 的计算效率,并通过数值算例对格式
性能进行了比较和验证。
3)提出和发展了最后一级为隐式的高精度 Runge-Kutta 格式。多级 Runge-Kutta 方
法具有精度高、稳定性好、存储量低等特点,但由于稳定性的要求,显式 Runge-Kutta
方法的时间步长受到一定的限制,为了增大时间步长,提高方法的稳定性,我们提出发
展最后一级为隐式的 Runge-Kutta 方法。与显式方法相比,最后一级为隐式的
Runge-Kutta 方法能够在更大的时间步长下推进并保持计算稳定和较低的耗散误差,因
此在对刚性问题或大规模长时间数值模拟时有助于提高计算效率,且由于只有一级需要
进行隐式迭代求解,因此比对角隐式方法具有更快的计算速度。结合稳定性及耗散色散
误差分析确定格式系数,本文给出了最后一级为隐式的三级三阶和四级四阶
Runge-Kutta 方法,并与空间激波捕捉格式相结合,应用于流体力学的数值模拟研究,
验证了方法的有效性。

Other Abstract

In the simulation of complex flows with large-scale or long-time integration, it is 
required that the numerical method is of high-order accuracy and high resolution, and also the 
computing process is always strong stable and high efficient. That means we should develop 
the high efficiency and high accuracy numerical methods for both the spatial and temporal 
discretizations for this kind of simulation. In recent years, the weighted essentially 
non-oscillatory(WENO) scheme is widely applied in CFD and continuously developed by 
researchers due to its excellent high-accuracy high-resolution ability for capturing complex 
flow characteristics. For the temporal integration, the Runge-Kutta method, which is well 
known as a classical method for ordinary differential equation, is usually used in CFD due to 
its high-accuracy, good-stability and low-storage. In this paper, we propose several methods 
to improve the performance of the WENO schemes and the Runge-Kutta method.
1)A shared-weights WENO method for solving the Euler equations is proposed. The
shared weights are calculated by using the energy flux components of Steger-Warming split 
method. Since the weights are shared by all flux components, the total computing efficiency 
is greatly improved. The variable used to calculate the shared weights plays an important role 
in the property of the WENO scheme, an inappropriate selection may result in an unstable 
process or low-resolution results. Numerical experiments showed that the shared-weights 
WENO method proposed in this paper has good robustness and low dissipation error.
2)An improved multi-step CRWENO scheme is proposed. Multi-step CRWENO scheme 
has spectral-like resolution as compact schemes in smooth regions, and meanwhile can obtain 
high accuracy and resolution around discontinuities. However, it has the same drawback of 
expensive computing cost as original multi-step WENO scheme. In this paper we extend the 
similar idea recently proposed by Zeng et al for improving the multi-step WENO scheme 
to the multi-step CRWENO scheme, and an improved multi-step CRWENO scheme is 
constructed. Numerical results of different schemes are compared and the performance of this 
new scheme is verified numerically.
3)This paper has developed a Runge-Kutta scheme with a last implicit stage. The 
Runge-Kutta method has advantages such as high accuracy, good stability and low storage. 
However, the time step of the explicit Runge-Kutta scheme is limited by the stability 
requirement. In order to increase the time step and stability, we have proposed a kind of
Runge-Kutta method with a last implicit stage. Compared to explicit RK scheme, the new 
method can take a larger time step for calculation while keeping stable and low dissipation 
error, this is helpful to improve the efficiency when it comes to simulation of stiff problems. 
In addition, only one implicit stage means its computation cost is apparently less than the 
diagonally implicit Runge-Kutta scheme. Using the analysis of stability and resolution 
property to determine coefficients, we have developed the third-order and fourth-order 
Runge-Kutta schemes with a last implicit stage. With a spatial shock-capturing scheme, the new RK schemes are applied in different hyperbolic equations. Numerical results have 
proved our theoretical analyses.

Language中文
Document Type学位论文
Identifierhttp://dspace.imech.ac.cn/handle/311007/79085
Collection中国科学院力学研究所
高温气体动力学国家重点实验室
Recommended Citation
GB/T 7714
唐荣. 欧拉方程的高效高精度时间及空间算法研究[D]. 北京. 中国科学院大学,2019.
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