﻿ 一维Euler方程的高分辨率有限体积格式 A High Resolution Finite Volume Scheme forOne Dimensional Euler Equations

International Journal of Fluid Dynamics
Vol.05 No.02(2017), Article ID:20924,13 pages
10.12677/IJFD.2017.52007

A High Resolution Finite Volume Scheme for One Dimensional Euler Equations

Yixin Kang1, Taofeng Xie2, Wei Gao1

1School of Mathematical Sciences, Inner Mongolia University, Hohhot Inner Mongolia

2School of Computer and Information Science, Inner Mongolia Medical University, Hohhot Inner Mongolia

Received: May 23rd, 2017; accepted: Jun. 9th, 2017; published: Jun. 12th, 2017

ABSTRACT

In this paper, a high resolution finite volume scheme is proposed to solve one dimensional Euler equations. We use the Hermite interpolation method to construct the present scheme. Two kinds of convection boundedness criteria TVD and CBC are combined to suppress nonphysical wiggles of linear schemes near discontinuities. Some typical test cases show that the present numerical scheme can solve one dimensional Euler equation with discontinuous initial profiles efficiently.

Keywords:Euler Equations, Hermite Interpolation, High Resolution, CBC, TVD

1内蒙古大学数学科学学院，内蒙古 呼和浩特

2内蒙古医科大学计算机信息学院，内蒙古 呼和浩特

1. 引言

2. 方法

2.1. Euler方程

Euler方程可以写成：

(1)

2.2. 对流有界性准则

(2)

(3)

Figure 1. Three neighboring mesh points and the mesh face

Table 1. The linear convection schemes and the normalized variable formations

(4)

(5)

(6)

(7)

(8)

NVF形式下的BAIR准则可以表达为

(9)

(10)

TVD准则被Swbey [7] 转化为限制函数：

2.3. 格式构造

1) 高分辨率格式的NVD线必须落在位于TVD和CBC-BAIR的区域内；

2) NVD线在NV坐标内必须通过点，且至少满足二阶精度差；

3) NVD线在NV坐标内满足，同时与格式具有相同的精度。

Figure 2. The regions of the TVD (shaded) and BAIR (hatched)

(11)

2.4. 时间项离散

(12)

(13)

3. 数值算例

3.1. 线性对流方程

(14)

Figure 3. The NV line of the HRFVM scheme in the BAIR and TVD region

3.1.1. 精确解算例

(15)

(16)

(17)

3.1.2. 间断解算例

(a) (b)

Figure 4. Comparison of numerical and exact results for the linear equations with nonsmooth initial distribution (a) CUI; (b) HRFVM

Table 2. Errors and orders for several selected schemes

3.2. 非线性方程

3.2.1.一维无粘性Burgers方程

(18)

3.2.2. Buckely-Leverett问题

(19)

Figure 5. Comparison of the numerical and exact solutions with one-dimen- sional inviscid Burgers equations

3.3. Euler方程

3.3.1. Lax激波管问题

Lax激波管问题，取黎曼初始条件

3.3.2. Shu-Osher 问题

Shu-Osher问题是一个典型的激波熵波相互作用的问题，激波同正弦波之间相互干扰，引发高频率振动，因而解的剖面图中既包含光滑解又包含波动解。

4. 总结

(a) (b)

Figure 6. The Buckley-Leverett equation (a) HRFVM and (b) GAMMA

Figure 7. The Lax problem of one-dimensional Euler equation: the computed solutions with the reference solutions: (a) Density; (b) Velocity; (c) Pressure; (d) Energy

Figure 8. The Shu-Osher problem with (a) 1000 cells and (b) 2000 cells

Euler方程的Lax问题和Shu-Osher问题，数值结果表明本文的数值格式能够有效地消除非物理振荡，并保持良好的间断逼近效果。

A High Resolution Finite Volume Scheme forOne Dimensional Euler Equations[J]. 流体动力学, 2017, 05(02): 56-68. http://dx.doi.org/10.12677/IJFD.2017.52007

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