﻿ 基于牛顿插值的高分辨率有限体积格式 A High-Resolution Finite Volume Scheme Based on Newtonian Interpolation

Vol.04 No.02(2015), Article ID:15228,11 pages
10.12677/AAM.2015.42020

A High-Resolution Finite Volume Scheme Based on Newtonian Interpolation

Wei Gao, Qing Zhang, Hong Li, Yang Liu

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

Received: Apr. 24th, 2015; accepted: May 10th, 2015; published: May 15th, 2015

ABSTRACT

Finite volume method plays an important role in fluid flow and heat transfer numerical calculation. How to eliminate unphysical oscillations caused by numerical solution of convection diffusion equation selecting discontinuity wave as the initial condition is a key task for studying finite volume method. New high-resolution schemes were constructed by Newton interpolation polynomial based on convection boundness criterion (CBC). Classic test cases demonstrated that the present numerical scheme possesses high resolution and good stability for high gradient and discontinuous solution.

Keywords:Convection Diffusion Equation, Newton Interpolation Polynomial, Convection Boundness Criterion (CBC), Total Variation Diminishing (TVD)

1. 引言

Figure 1. Three neighboring mesh points and the mesh face

2. 对流有界性准则

(2.1)

Table 1. The linear convection schemes and the NV formulations

3. 格式的构造

1) 由图2可知

2)，参数的范围为而且

4. 时间离散格式

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

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

5. 数值结果及讨论

5.1. 一维线性对流方程

(5.1)

5.1.1. 情形1

5.1.2. 情形2

Table 2. Errors and orders for several selected schemes

5.1.3. 情形3

5.2. 一维非线性Burgers方程

5.2.1. 情形1

(a) CFL = 0.2, T = 0.5, N = 800 (b) CFL = 0.5, T = 0.5, N = 800

Figure 4. Comparison of MUSCL and NPUS scheme for the linear equation with nonsmooth initial distribution

(a) CFL = 0.2, T = 1.0, N = 800 (b) CFL = 0.5, T = 1.0, N = 800

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

Figure 6. One-dimensional inviscid Burgers equation: comparison of the numerical solutions by the NPUS with 400 cells at CEL = 0.5 with the exact solution

5.2.2. 情形2

5.3. 一维非线性Buckely-Leverett方程

5.4. 二维线性对流方程

(a) CFL = 0.2, T = 1.0, N = 400 (b) CFL = 0.5, T = 1.0, N = 400 (c) CFL = 0.2, T = 2.0, N = 400 (d) CFL = 0.5, T = 2.0, N = 400

Figure 7. One-dimensional inviscid Burgers equation with nonsmooth initial distribution: comparison of the numerical solutions by the NPUS with the exact solution

Figure 8. Comparison of numerical and exact results of the Buckley-Leverett equation

(a) 1-D case: x = 0 (b) 2-D case

Figure 9. Exact and numerical solutions of Doswell frontogenesis at t = 4. The meshes of 200 × 200 are used

Figure 10. The numerical solution of Doswell frontogenesis in 3D view

6. 总结

A High-Resolution Finite Volume Scheme Based on Newtonian Interpolation. 应用数学进展,02,150-161. doi: 10.12677/AAM.2015.42020

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