基于四阶CWENO重构的熵相容格式研究

发布时间：2018-12-13 08:37

【摘要】：随着计算机科学技术日新月异的发展,运用数值计算方法求解计算流体力学中双曲守恒律方程也变得尤为重要。近年来,为保证数值计算方法所求的双曲守恒律的解具有物理意义,根据热力学第二定律,发展了满足熵稳定条件的一些数值求解格式,如本文中研究的熵稳定和熵相容格式,可以避免“膨胀激波”、“色散效应”等非物理现象,应用前景良好。鉴于此,本文从双曲守恒律的物理背景出发,研究熵相容格式的思想理论和构造方法,并在现有的熵相容格式基础上,通过在单元交界面处进行左右状态值的四阶CWENO型重构,设计新型熵相容格式,提高数值格式的精度。并通过Burgers方程和Euler方程进行数值实验,验证新格式在捕捉间断时具有的良好特性,如高精度、鲁棒性、无震荡性等。本文所做的主要工作如下:(1)从求解双曲守恒律方程的熵守恒格式出发重点详述了熵稳定、熵相容及高分辨率熵相容格式的发展,并通过各个格式求解一维无粘Burgers方程间断初值问题,验证各个格式的特点及其捕捉间断的不同效果。(2)通过对计算单元交界面左右值进行四阶CWENO重构,将重构后的左右状态值代入熵相容格式中,结合LeFloch提出的四阶熵守恒格式,构造高精度熵相容格式。并通过对Burgers方程的求解,对比其与原熵相容格式的特点,说明新格式的无震荡性、捕捉间断的有效性。(3)构造求解Euler方程的高精度熵相容数值格式,通过对Euler方程的求解,验证新格式的特点,体现其无震荡性、高精度、高分辨率、鲁棒性等特点。
[Abstract]:With the rapid development of computer science and technology, it is very important to solve hyperbolic conservation law equations in computational fluid dynamics by numerical method. In recent years, in order to ensure that the solution of hyperbolic conservation law obtained by numerical method has physical significance, according to the second law of thermodynamics, some numerical solutions satisfying the condition of entropy stability are developed, such as entropy stability scheme and entropy compatible scheme studied in this paper. It can avoid non-physical phenomena such as "expansion shock wave" and "dispersion effect", and has a good prospect in application. In view of this, from the physical background of hyperbolic conservation law, this paper studies the thought theory and construction method of entropy compatible scheme. Based on the existing entropy compatible scheme, the fourth order CWENO type reconstruction of the left and right state values at the interface of the unit is carried out. A new entropy compatible scheme is designed to improve the accuracy of numerical scheme. Through the numerical experiments of Burgers equation and Euler equation, it is verified that the new scheme has good characteristics in capturing discontinuity, such as high precision, robustness, non-oscillation and so on. The main work of this paper is as follows: (1) starting from the entropy conservation scheme for solving hyperbolic conservation law equation, the development of entropy stability, entropy compatibility and high resolution entropy compatible schemes are described in detail. The discontinuous initial value problem of one-dimensional inviscid Burgers equation is solved by each scheme, and the characteristics of each scheme and the different effects of capturing the discontinuity are verified. (2) the fourth order CWENO reconstruction is carried out on the left and right values of the interface of the computing unit. The reconstructed left and right state values are substituted into the entropy compatible scheme and combined with the fourth order entropy conservation scheme proposed by LeFloch to construct the high precision entropy compatible scheme. By solving the Burgers equation, comparing its characteristics with the original entropy compatible scheme, the new scheme is shown to be non-oscillating, and the effectiveness of capturing the discontinuity is demonstrated. (3) the high-precision entropy consistent numerical scheme for solving the Euler equation is constructed, and the solution of the Euler equation is obtained. Verify the characteristics of the new format, reflect its non-oscillating, high precision, high resolution, robustness and so on.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82

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