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Analysis on the Influence of the Expansion Coefficient in the Non-Uniform Grids for the Accuracy of Compact Finite Difference Schemes
DOI: 10.12677/PM.2016.64047, PDF, HTML, XML, 下载: 1,406  浏览: 3,292  国家自然科学基金支持

Abstract: In the numerical simulation of problems with large gradient and boundary layer, large amount of calculation or large calculation error will occur if uniform grids are used. Non-uniform grids can decrease calculation error greatly while keep the same computational expense. The non- uniformity of the non-uniform grids can be controlled by an expansion coefficient, which affects the accuracy of a scheme. In this paper, analysis has been presented for the effect of the expan-sion coefficient to the numerical results of one dimensional convective-diffusion equation by using compact difference scheme under non-uniform grids. Two numerical examples are given and it is indicated that optimal value of the expansion coefficient may exist. The computation accuracy can be increased greatly by choosing reasonable expansion coefficient. Moreover, comparisons among compact difference scheme on both non-uniform grids and uniform grids and Crank-Nicolson schemes show that the compact difference on non-uniform grids can be used to solve the problem of large gradient and boundary layer with high accuracy.

 [1] Tian, Z.F. and Feng, X.F. (2000) A New Explicit Method with Exponential-Type for the Convection-Diffusion Equation. Chinese Journal of Engineering Mathematics, 17, 65-69. [2] 葛永斌, 田振夫, 吴文权. 含源项非定常对流扩散方程的高精度紧致隐式差分方法[J]. 水动力学研究与进展(A辑), 2006(21): 619-625. [3] 开依沙尔•热合曼, 阿孜古丽•牙生, 祖丽皮耶•如孜. 求解一维对流扩散方程的高精度紧致差分格式[J]. 佳木斯大学学报, 2014, 32(1): 135-138. [4] 赵飞, 蔡志权, 葛永斌. 一维非定常对流扩散方程的有理型高阶紧致差分格式[J]. 江西师范大学学报, 2014, 38(4): 413-418. [5] Tian, Z.F. and Yu, P.X. (2011) A High-Order Exponential Scheme for Solving One-Dimensional Unsteady Convection-Diffusion Equations. Journal of Computational and Applied Mathematics, 235, 2477-2491. http://dx.doi.org/10.1016/j.cam.2010.11.001 [6] Anderson, J.D. (1995) Computational Fluids Dynamics. McGraw-Hill Education, New York. [7] 孙建安, 贾伟, 吴广智. 一种非均匀网格上的高精度紧致差分格式[J]. 西北师范大学学报, 2014, 50(4): 31-35. [8] 田芳, 田振夫. 非均匀网格上求解对流扩散方程问题的高阶紧致差分方法[J]. 宁夏大学学报, 2009, 30(3): 209- 212. [9] 黄雪芳, 郭锐, 葛永斌. 一维非定常对流扩散方程非均匀网格上的高精度紧致差分格式[J]. 工程数学学报, 2014, 31(3): 371-380. [10] Akbar, M. and Mehdi, D. (2010) High-Order Compact Solution of the One-Dimensional Heat and Advection-Diffusion Equation. Applied Mathematical Modelling, 34, 3071-3084. http://dx.doi.org/10.1016/j.apm.2010.01.013 [11] Zerroukat, M., Djidjel, K. and Charafi, A. (2000) Explicit and Implicit Meshless Methods for Linear Advection Diffusion-Type Partial Differential Equations. International Journal for Numerical Methods in Engineering, 48, 9-35. http://dx.doi.org/10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3