﻿ 对流占优问题的一种修正CUI格式 A Modified CUI Scheme for Convection-Dominated Equations

Vol.05 No.04(2016), Article ID:19086,12 pages
10.12677/AAM.2016.54083

A Modified CUI Scheme for Convection-Dominated Equations

Na Lv1, Taofeng Xie2, Wei Gao1

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

2College of Computer and Information, Inner Mongolia Medical University, Hohhot Inner Mongolia

Received: Nov. 9th, 2016; accepted: Nov. 24th, 2016; published: Nov. 29th, 2016

ABSTRACT

In this paper, a modified CUI scheme is presented for discretizing the convection term. Coupled with Herimite interpolation, CBC (Convection Boundedness Criterion) and TVD (Total Variational Diminishing Constraint) are applied to suppress numerical oscillations. Typical test cases demonstrate that the present scheme possesses the boundedness of convection and high accuracy.

Keywords:CUI Scheme, Hermite Interpolation Polynomial, CBC/TVD, mCUI Scheme

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. 建立数值格式

2.1. 高阶格式

(1)

(2)

Figure 1. Three neighboring mesh points and the mesh face

Table 1. The linear convection schemes and the NV formulations

2.2. 对流项的离散

2.3. 时间离散

3. 数值结果及讨论

3.1. 一维线性对流方程

(3)

3.1.1. 情形1

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

Figure 3. The illustration of the NV line of the mCUI scheme in the BAIR region

mCUI格式、SMART [7] 格式、MUSCL [14] 格式在等距剖分下的数值解，分别取20、40、80、160、320。计算格式的误差、误差，同时算出数值精度阶(如表2)，计算公式如下

Table 2. Errors and orders for several selected schemes

3.1.2. 情形2

3.1.3. 情形3

3.2. 一维非线性Burgers方程

(4)

Figure 4. Comparison of numerical and exact results for the discontinuous initial condition

(a) (b)

Figure 5.Comparison of MUSCL and mCUI scheme for the linear equation with nonsmooth initial distribution

3.2.1. 情形1

3.2.2. 情形2

，即对于无粘Burgers方程，选取以下的初值

3.3. 二维线性对流方程

(a) (b) (c) (d)

Figure 6. Numerical results for Burgers equation with different viscosity coefficients, at six selected time instants, (a) , (b) , (c) , (d)

(a) (b)

Figure 7. Comparison between the exact and numerical solutions for the inviscid Burgers equation at selected time instants, (a) t = 1.0, (b) t = 2.0

(a) (b)(c)

Figure 8. Exact and numerical solutions of Doswell and 2D, 3D image

3.4. 二维无粘性Burgers方程

mCUI格式可以解决二维无粘性的Burgers方程

(a) (b)

Figure 9. The two-dimensional inviscid Burgers equation; computed solutions at two selected time instants. (a) t = 0.1 and (b) t = 0.6

4. 结论

A Modified CUI Scheme for Convection-Dominated Equations[J]. 应用数学进展, 2016, 05(04): 716-727. http://dx.doi.org/10.12677/AAM.2016.54083

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