基于三阶CWENO重构的高精度高分辨率熵相容格式研究

发布时间：2018-05-07 17:12

本文选题：双曲型守恒律方程 + 熵守恒/熵稳定/熵相容　；参考：《长安大学》2015年硕士论文

【摘要】：双曲守恒律方程是流体力学中重要的物理模型,因此研究能对它准确求解的数值方法是非常重要的。近年来,根据守恒律的基础理论和热力学第二定律,即熵稳定条件发展起来的熵守恒、熵稳定和熵相容格式对双曲型守恒律方程的求解表现出了重要的意义。本文就这三种格式的主要思想和构造方法进行了适当的研究,并根据Burgers方程的这三种格式的构造方法对浅水波方程的熵守恒、熵稳定和熵相容格式进行了具体的构造。在此基础之上,本文运用三阶CWENO重构和添加适当的限制器的方法构造出了一种新的高精度高分辨率熵相容格式,并通过数值模拟验证了格式的有效性。主要工作如下:(1)详细研究了熵守恒、熵稳定和熵相容格式的构造思想和特点。熵守恒格式是二阶精度的,但没有耗散机制,因此在间断处会出现振荡。熵稳定格式是对熵守恒格式的改进,给其加入耗散机制以减小振荡。熵相容格式是在熵稳定格式上添加了一个较小的耗散项来满足与实际物理意义上的耗散量相容的要求,从而达到消除或减小振荡的效果。(2)为了更准确地计算问题的数值解,本文通过三阶CWENO重构和给熵相容格式添加适当的通量限制器,构造了一种高精度高分辨率熵相容格式。该格式不仅避免了非物理现象的出现而且能够精确地捕捉稀疏波、激波等,在求解双曲型守恒律方程的过程中表现出了不容忽视的优越性。(3)通过对Burgers方程的格式构造及数值模拟来验证新构造的高精度高分辨率熵相容格式的有效性。(4)针对浅水波方程,在Fjordholm提出的熵守恒格式基础之上,构造了熵稳定、熵相容、高分辨率熵相容、高精度高分辨率熵相容格式。然后,通过数值模拟,比较各格式的有效性,验证高精度高分辨率熵相容格式的优越性。
[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.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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