﻿ 三角形网格下对流扩散方程的无震荡格式 A Non-Oscillatory Scheme for Convection Diffusion Equations on Triangular Meshes

International Journal of Fluid Dynamics
Vol.06 No.02(2018), Article ID:25406,10 pages
10.12677/IJFD.2018.62004

A Non-Oscillatory Scheme for Convection Diffusion Equations on Triangular Meshes

Juan Zhao, Wei Gao*

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

Received: May 24th, 2018; accepted: Jun. 6th, 2018; published: Jun. 14th, 2018

ABSTRACT

Construction of oscillation-free schemes on unstructured meshes plays a key role on numerical solutions to convection dominated problems. A new NVSF (Normalized Variable and Space Formulation) scheme is presented on triangular meshes. The typical test cases show that the present scheme can suppress unphysical oscillations and possess good accuracy.

Keywords:Convection Diffusion Equation, Triangular Meshes, Non-Oscillatory Scheme

1. 引言

2. 非结构网格下格式的构造

2.1. 非一致网格下的NVSF的正则化

$\stackrel{^}{\varphi }=\frac{\varphi -{\varphi }_{U}}{{\varphi }_{D}-{\varphi }_{U}},\text{\hspace{0.17em}}\stackrel{^}{x}=\frac{x-{x}_{U}}{{x}_{D}-{x}_{U}}$

2.2. 对流项有界性准则

$\left\{\begin{array}{l}{\stackrel{^}{\varphi }}_{C}<{\stackrel{^}{\varphi }}_{f}=f\left({\stackrel{^}{\varphi }}_{C}\right)<1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<{\stackrel{^}{\varphi }}_{C}<1\hfill \\ {\stackrel{^}{\varphi }}_{f}={\stackrel{^}{\varphi }}_{C},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{\varphi }}_{C}\le 0\hfill \\ {\stackrel{^}{\varphi }}_{f}={\stackrel{^}{\varphi }}_{C},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{^}{\varphi }}_{C}\ge 1\hfill \end{array}$ (1)

Figure 1. Three neighboring mesh points and the mesh face on unstructured meshes

Figure 2. The relationship between variables before the regularization

Figure 3. The relationship between variables after the regularization

2.3. 新格式的构造

${\stackrel{^}{\phi }}_{f}=a{\stackrel{^}{\phi }}_{C}^{3}+b{\stackrel{^}{\phi }}_{C}^{2}+c{\stackrel{^}{\phi }}_{C}+d$ (2)

Figure 4. Normalized Variable Diagram (NVD) for several linear schemes formulated using NVSF

$\left\{\begin{array}{l}{\stackrel{^}{\phi }}_{f}\left(0\right)=0\\ {\stackrel{^}{\phi }}_{f}\left({\stackrel{^}{x}}_{c}\right)={\stackrel{^}{x}}_{f}\\ {\stackrel{^}{\phi }}_{f}\left(1\right)=1\end{array}$ (3)

${{\stackrel{^}{\phi }}^{\prime }}_{f}\left({\stackrel{^}{x}}_{c}\right)=\frac{{\stackrel{^}{x}}_{f}\left({\stackrel{^}{x}}_{f}-1\right)}{{\stackrel{^}{x}}_{c}\left({\stackrel{^}{x}}_{c}-1\right)}$ (4)

$\left\{\begin{array}{l}a=\frac{{\left({\stackrel{^}{x}}_{c}-{\stackrel{^}{x}}_{f}\right)}^{2}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}\\ b=\frac{3{\stackrel{^}{x}}_{c}^{2}{\stackrel{^}{x}}_{f}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}-2{\stackrel{^}{x}}_{c}^{3}-{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}^{2}-{\stackrel{^}{x}}_{f}^{2}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}\\ c=\frac{{\stackrel{^}{x}}_{c}^{4}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}^{2}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}-3{\stackrel{^}{x}}_{c}^{2}{\stackrel{^}{x}}_{f}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}\\ d=0\end{array}$

$\begin{array}{c}{\stackrel{^}{\phi }}_{f}=\frac{{\left({\stackrel{^}{x}}_{c}-{\stackrel{^}{x}}_{f}\right)}^{2}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}{\stackrel{^}{\phi }}_{C}^{3}+\frac{3{\stackrel{^}{x}}_{c}^{2}{\stackrel{^}{x}}_{f}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}-2{\stackrel{^}{x}}_{c}^{3}-{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}^{2}-{\stackrel{^}{x}}_{f}^{2}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}{\stackrel{^}{\phi }}_{C}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\stackrel{^}{x}}_{c}^{4}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}^{2}+{\stackrel{^}{x}}_{c}{\stackrel{^}{x}}_{f}-3{\stackrel{^}{x}}_{c}^{2}{\stackrel{^}{x}}_{f}}{{\stackrel{^}{x}}_{c}^{2}{\left({\stackrel{^}{x}}_{c}-1\right)}^{2}}{\stackrel{^}{\phi }}_{C}\end{array}$

${\stackrel{^}{\varphi }}_{f}={\stackrel{^}{\varphi }}_{C}^{3}-\frac{5}{2}{\stackrel{^}{\varphi }}_{C}^{2}+\frac{5}{2}{\stackrel{^}{\varphi }}_{C}$ (5)

3. 三角形剖分的变量关系

Figure 5. New Format Curve and CBC Guidelines Area (area of dotted line)

Figure 6. The relationship between variables of the triangle mesh

${\phi }_{U}=\frac{\sum _{i=1}^{n}\frac{{\phi }_{i}}{{d}_{i}}}{\sum _{i=1}^{n}\frac{1}{{d}_{i}}}$

4. 时间的离散

$\left\{\begin{array}{l}{\varphi }^{\left(1\right)}={\varphi }^{n}+L\left({\varphi }^{\left(1\right)}\right)\hfill \\ {\varphi }^{\left(2\right)}=\frac{3}{4}{\varphi }^{n}+\frac{1}{4}\left[{\varphi }^{\left(1\right)}+L\left({\varphi }^{\left(1\right)}\right)\right]\hfill \\ {\varphi }^{n+1}=\frac{1}{3}{\varphi }^{n}+\frac{2}{3}\left[{\varphi }^{n}+L\left({\varphi }^{n}\right)\right]\hfill \end{array}$

5. 数值算例

5.1. 线性对流方程

$\frac{\partial \varphi }{\partial t}+\alpha \frac{\partial \varphi }{\partial x}=0$ , $x\in \left[a,b\right]$ , $t>0$ (6)

5.1.1. 情形1

$\text{order}=\frac{\mathrm{ln}{E}_{N}/{E}_{2N}}{\mathrm{ln}2}$

5.1.2. 情形2

$\varphi \left(x,0\right)=\left\{\begin{array}{c}1,\\ 0,\end{array}\begin{array}{c}x\le 0\\ x\ge 0\end{array}$

5.2. 二维线性对流方程

Table 1. Errors and orders for several selected schemes

Figure 7. Comparison of numerical and exact results for the new condition

$\frac{\partial \varphi }{\partial t}+\frac{\partial \left(a\varphi \right)}{\partial x}+\frac{\partial \left(b\varphi \right)}{\partial y}=0$

${\nu }_{t}\left(r\right)=\frac{1}{{\nu }_{\mathrm{max}}}\frac{\mathrm{tanh}\left(r\right)}{{\mathrm{cosh}}^{2}\left(r\right)}$

$a\left(x,y\right)=-\frac{y-{x}_{0}}{r}{\nu }_{t}\left(r\right)$ , $b\left(x,y\right)=\frac{x-{y}_{0}}{r}{\nu }_{t}\left(r\right)$

$\varphi \left(x,y,t\right)=\mathrm{tanh}\left[\frac{y-{x}_{0}}{\delta }\mathrm{cos}\left(\omega t\right)-\frac{x-{y}_{0}}{\delta }\mathrm{sin}\left(\omega t\right)\right]$

6. 结论

Figure 8. Triangulation diagram of EasyMesh

(a) (b)

Figure 9. Exact and numerical solutions of Doswell

A Non-Oscillatory Scheme for Convection Diffusion Equations on Triangular Meshes[J]. 流体动力学, 2018, 06(02): 23-32. https://doi.org/10.12677/IJFD.2018.62004

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