本文選題：雙曲型守恒律方程 + 熵守恒/熵穩(wěn)定/熵相容　； 參考：《長安大學(xué)》2015年碩士論文

【摘要】：雙曲守恒律方程是流體力學(xué)中重要的物理模型,因此研究能對(duì)它準(zhǔn)確求解的數(shù)值方法是非常重要的。近年來,根據(jù)守恒律的基礎(chǔ)理論和熱力學(xué)第二定律,即熵穩(wěn)定條件發(fā)展起來的熵守恒、熵穩(wěn)定和熵相容格式對(duì)雙曲型守恒律方程的求解表現(xiàn)出了重要的意義。本文就這三種格式的主要思想和構(gòu)造方法進(jìn)行了適當(dāng)?shù)难芯?并根據(jù)Burgers方程的這三種格式的構(gòu)造方法對(duì)淺水波方程的熵守恒、熵穩(wěn)定和熵相容格式進(jìn)行了具體的構(gòu)造。在此基礎(chǔ)之上,本文運(yùn)用三階CWENO重構(gòu)和添加適當(dāng)?shù)南拗破鞯姆椒?gòu)造出了一種新的高精度高分辨率熵相容格式,并通過數(shù)值模擬驗(yàn)證了格式的有效性。主要工作如下:(1)詳細(xì)研究了熵守恒、熵穩(wěn)定和熵相容格式的構(gòu)造思想和特點(diǎn)。熵守恒格式是二階精度的,但沒有耗散機(jī)制,因此在間斷處會(huì)出現(xiàn)振蕩。熵穩(wěn)定格式是對(duì)熵守恒格式的改進(jìn),給其加入耗散機(jī)制以減小振蕩。熵相容格式是在熵穩(wěn)定格式上添加了一個(gè)較小的耗散項(xiàng)來滿足與實(shí)際物理意義上的耗散量相容的要求,從而達(dá)到消除或減小振蕩的效果。(2)為了更準(zhǔn)確地計(jì)算問題的數(shù)值解,本文通過三階CWENO重構(gòu)和給熵相容格式添加適當(dāng)?shù)耐肯拗破?構(gòu)造了一種高精度高分辨率熵相容格式。該格式不僅避免了非物理現(xiàn)象的出現(xiàn)而且能夠精確地捕捉稀疏波、激波等,在求解雙曲型守恒律方程的過程中表現(xiàn)出了不容忽視的優(yōu)越性。(3)通過對(duì)Burgers方程的格式構(gòu)造及數(shù)值模擬來驗(yàn)證新構(gòu)造的高精度高分辨率熵相容格式的有效性。(4)針對(duì)淺水波方程,在Fjordholm提出的熵守恒格式基礎(chǔ)之上,構(gòu)造了熵穩(wěn)定、熵相容、高分辨率熵相容、高精度高分辨率熵相容格式。然后,通過數(shù)值模擬,比較各格式的有效性,驗(yàn)證高精度高分辨率熵相容格式的優(yōu)越性。
[Abstract]:Hyperbolic conservation law equation is an important physical model in fluid mechanics, so it is very important to study the numerical method which can be solved accurately. In recent years, according to the basic theory of conservation law and the second law of thermodynamics, namely entropy conservation, entropy stability and entropy compatible scheme developed from entropy stability condition, it is very important to solve hyperbolic conservation law equation. In this paper, the main ideas and construction methods of these three schemes are studied properly, and the conservation of entropy, entropy stability and entropy compatible scheme of shallow water wave equations are constructed according to the construction methods of these three schemes of Burgers equation. On this basis, a new high-resolution entropy compatible scheme with high accuracy and high resolution is constructed by using third-order CWENO reconstruction and adding appropriate limiter, and the validity of the scheme is verified by numerical simulation. The main work is as follows: (1) We study in detail the construction ideas and characteristics of entropy conservation, entropy stability and entropy compatible schemes. The entropy conservation scheme is of second-order accuracy, but there is no dissipative mechanism, so oscillations occur at the discontinuity. The entropy stable scheme is an improvement to the entropy conservation scheme, and a dissipative mechanism is added to it to reduce the oscillation. The entropy compatible scheme adds a small dissipation term to the entropy stable scheme to meet the requirement of compatibility with the actual physical dissipation, thus achieving the effect of eliminating or reducing the oscillation.) in order to calculate the numerical solution of the problem more accurately, In this paper, a high precision and high resolution entropy compatible scheme is constructed by using third-order CWENO reconstruction and adding appropriate flux limiter to entropy compatible scheme. This scheme not only avoids the appearance of non-physical phenomena, but also can accurately capture sparse waves, shock waves, etc. In the process of solving the hyperbolic conservation law equation, it shows the superiority that can not be ignored.) the validity of the new high precision and high resolution entropy compatible scheme is verified by the scheme construction and numerical simulation of the Burgers equation. Based on the entropy conservation scheme proposed by Fjordholm, the entropy stability, entropy compatibility, high resolution entropy compatibility and high resolution entropy compatibility schemes are constructed. Then, through numerical simulation, the validity of each scheme is compared to verify the superiority of high precision and high resolution entropy compatible scheme.
【學(xué)位授予單位】：長安大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：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(xué);2015年

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本文編號(hào)：1857763