【摘要】：隨著計算機科學技術(shù)日新月異的發(fā)展,運用數(shù)值計算方法求解計算流體力學中雙曲守恒律方程也變得尤為重要。近年來,為保證數(shù)值計算方法所求的雙曲守恒律的解具有物理意義,根據(jù)熱力學第二定律,發(fā)展了滿足熵穩(wěn)定條件的一些數(shù)值求解格式,如本文中研究的熵穩(wěn)定和熵相容格式,可以避免“膨脹激波”、“色散效應”等非物理現(xiàn)象,應用前景良好。鑒于此,本文從雙曲守恒律的物理背景出發(fā),研究熵相容格式的思想理論和構(gòu)造方法,并在現(xiàn)有的熵相容格式基礎上,通過在單元交界面處進行左右狀態(tài)值的四階CWENO型重構(gòu),設計新型熵相容格式,提高數(shù)值格式的精度。并通過Burgers方程和Euler方程進行數(shù)值實驗,驗證新格式在捕捉間斷時具有的良好特性,如高精度、魯棒性、無震蕩性等。本文所做的主要工作如下:(1)從求解雙曲守恒律方程的熵守恒格式出發(fā)重點詳述了熵穩(wěn)定、熵相容及高分辨率熵相容格式的發(fā)展,并通過各個格式求解一維無粘Burgers方程間斷初值問題,驗證各個格式的特點及其捕捉間斷的不同效果。(2)通過對計算單元交界面左右值進行四階CWENO重構(gòu),將重構(gòu)后的左右狀態(tài)值代入熵相容格式中,結(jié)合LeFloch提出的四階熵守恒格式,構(gòu)造高精度熵相容格式。并通過對Burgers方程的求解,對比其與原熵相容格式的特點,說明新格式的無震蕩性、捕捉間斷的有效性。(3)構(gòu)造求解Euler方程的高精度熵相容數(shù)值格式,通過對Euler方程的求解,驗證新格式的特點,體現(xiàn)其無震蕩性、高精度、高分辨率、魯棒性等特點。
[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.
【學位授予單位】：長安大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O241.82

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1 楊婷;基于三階CWENO重構(gòu)的高精度高分辨率熵相容格式研究[D];長安大學;2015年

2 顏克清;基于四階CWENO重構(gòu)的熵相容格式研究[D];長安大學;2016年

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本文編號：2376254