A Modified Finite Volume Approximation of 2-Dimensional Diffusion Equation With Discontinuous Coefficients

XU Xiao-lei; FENG Xiu-fang

doi:10.3879/j.issn.1000-0887.2014.02.003

Volume 35 Issue 2

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2014 > 35(2): 130-147

XU Xiao-lei, FENG Xiu-fang. A Modified Finite Volume Approximation of 2-Dimensional Diffusion Equation With Discontinuous Coefficients[J]. Applied Mathematics and Mechanics, 2014, 35(2): 130-147. doi: 10.3879/j.issn.1000-0887.2014.02.003

Citation:

PDF( 2396 KB)

A Modified Finite Volume Approximation of 2-Dimensional Diffusion Equation With Discontinuous Coefficients

doi: 10.3879/j.issn.1000-0887.2014.02.003

School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, P.R.China

Funds: The National Natural Science Foundation of China(11161036); The Key Project of Chinese Ministry of Education(209134)

Received Date: 2013-08-16
Rev Recd Date: 2013-10-31
Publish Date: 2014-02-15

Abstract

Abstract

A new modified finite volume method was presented to solve the 2-dimensional diffusion equation. Through improvement of the methods for solving the flux function and harmonic average coefficient, a new difference scheme was obtained for the diffusion equation with discontinuous coefficients. This scheme was an implicit difference scheme and was unconditionally stable. Subsequent numerical tests show that the presented method is more accurate than the classical finite volume method.
- diffusion equation with discontinuous coefficient,
- finite volume method,
- interface problem

FullText(HTML)

References(14)

References

[1]	Mohebbi A, Dehghan M. High order compact solution of the one dimensional heat and advection-diffusion equations[J]. Appl Math Modell,2010,34(10): 3071-3084.
[2]	王彩华. 一维对流扩散方程的一类高精度紧致差分格式[J]. 水动力学研究与进展, 2004,19(5): 654-663.(WANG Cai-hua. A class of high-order compact difference scheme for a convection-diffusion equation[J]. Journal of Hydrodynamics,2004,19(5): 654-663.(in Chinese))
[3]	田芳, 田振夫. 二维对流扩散方程非均匀网格上的高阶紧致差分方法[J]. 水动力学研究与进展, A辑, 2008,23(5): 475-483.(TIAN Fang, TIAN Zhen-fu. A high-order compact difference method for 2D convection-diffusion equation on non-uniform grid[J]. Chinese Journal of Hydrodynamics, Ser A,2008,23(5): 475-483.(in Chinese))
[4]	葛永斌, 田振夫. 求解扩散方程的一种高精度隐式差分方法[J]. 上海理工大学学报, 2005,27(2): 107-110.(GE Yong-bin, TIAN Zhen-fu. High-order implicit difference method for the diffusion equation[J]. Journal of University of Shanghai for Science and Technology,2005,27(2): 107-110.(in Chinese))
[5]	Evans D J, Abdullah A R. Alternating group explicit methods for the diffusion equations[J]. Appl Math Modelling，1985,9(3): 201-206.
[6]	张宝琳. 求解扩散方程的交替分段显-隐式方法[J]. 数值计算与计算机应用, 1991,12(4): 245-253.(ZHANG Bao-lin. An alternating segment explicit-implicit method for the diffusion equation[J]. Journal of Numerical Methods and Computer Applications,1991,12(4): 245-253.(in Chinese))
[7]	田振夫, 冯秀芳. 对流扩散方程的一种新的显式方法[J]. 工程数学学报, 2000,17(4): 65-69.(TIAN Zhen-fu, FENG Xiu-fang. A new explicit method with exponential-type for the convection-diffusion equation[J]. Journal of Engineering Mathematics,2000,17(4): 65-69.(in Chinese))
[8]	汪继文, 窦红. 求解对流扩散方程的一种高效的有限体积法[J]. 应用力学学报, 2008,25(3): 480-483.(WANG Ji-wen, DOU Hong. High-efficient finite volume method for advection-diffusion equations[J]. Chinese Journal of Applied Mechanics,2008,25(3): 480-483.(in Chinese))
[9]	高智, 柏威. 对流扩散方程的摄动有限体积(PFV)方法及讨论[J]. 力学学报, 2004,36(1): 88-93.(GAO Zhi, BAI Wei. Perturbational finite volume method for convective diffusion equation and discussion[J]. Acta Mechanica Sinica,2004,36(1): 88-93.(in Chinese))
[10]	宋淑红, 王双虎. 带间断扩散系数热传导方程的新型自适应数解法[J]. 应用数学学报, 2010,33(5): 942-959.(SONG Shu-hong, WANG Shuang-hu. A new adaptive numerical solver for heat conduction equation with discontinuous diffusion coefficient[J]. Acta Mathematical Applicatae Sinica,2010,33(5): 942-959.(in Chinese))
[11]	宋淑红, 王双虎. 带间断扩散系数热传导方程的高精度数值模拟方法研究[J]. 应用数学学报, 2011,34(2): 229-239.(SONG Shu-hong, WANG Shuang-hu. High accuracy numerical method for heat conduction equation with discontinuous diffusion coefficient[J]. Acta Mathematical Applicatae Sinica,2011,34(2): 229-239. (in Chinese))
[12]	赵强, 袁光伟, 董志伟. 基于节点重构的扩散方程有限体积格式[J]. 计算物理, 2012,29(1): 1-9.(ZHAO Qiang, YUAN Guang-wei, DONG Zhi-wei. Finite volume scheme for diffusion equation with vertex reconstruction[J]. Chinese Journal of Computational Physics,2012,29(1): 1-9.(in Chinese))
[13]	Krylov N V. Nonlinear Elliptic and Parabolic Equations of Second Order [M]. Springer, 1987.
[14]	Ladyenskaja A O, Solonnikov V A, Ural’ceva N N. Linear and Quasi-Linear Equations of Parabolic Type [M]. Smith S trans. Translations of Mathematical Monographs, Vol23, 1968.

Relative Articles

[1]	NI Yunlin, ZHANG Xijiang, YU Jiangmei, SHEN Liangduo. Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed[J]. Applied Mathematics and Mechanics, 2024, 45(5): 606-621. doi: 10.21656/1000-0887.440255
[2]	ZHANG Shangyuan, NIE Yufeng, LI Yiqiang. The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations[J]. Applied Mathematics and Mechanics, 2022, 43(6): 608-618. doi: 10.21656/1000-0887.420295
[3]	CHEN Yafei, ZHENG Yunying. A Discontinuous Galerkin FEM for 2D Navier-Stokes Equations of Incompressible Viscous Fluids[J]. Applied Mathematics and Mechanics, 2020, 41(8): 844-852. doi: 10.21656/1000-0887.400379
[4]	QIN Yan-mei, LI Hui, FENG Min-fu. A Local Stabilized Nonconforming Finite Element Method for the Optimal Control of Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2016, 37(8): 842-855. doi: 10.21656/1000-0887.370137
[5]	ZHOU Huan-lin, XU Xing-sheng, LI Xiu-li, CHEN Hao-long. Identification of Temperature-Dependent Thermal Conductivity for 2-D Transient Heat Conduction Problems[J]. Applied Mathematics and Mechanics, 2014, 35(12): 1341-1351. doi: 10.3879/j.issn.1000-0887.2014.12.006
[6]	TENG Fei, LUO Zhen-dong. A Reduced-Order Stabilized CNFVE Extrapolating Model for Non-Stationary Stokes Equations[J]. Applied Mathematics and Mechanics, 2014, 35(9): 986-1001. doi: 10.3879/j.issn.1000-0887.2014.09.005
[7]	CHEN Ye-fei, LI Wen-cheng, DENG Zi-chen>. Discontinuous Galerkin Finite Element Method Based on Rosenbrock-Type Exponential Integrator[J]. Applied Mathematics and Mechanics, 2013, 34(7): 697-703. doi: 10.3879/j.issn.1000-0887.2013.07.004
[8]	SHI Dong-yang, XU Chao. Anisotropic Nonconforming Crouzeix-Raviart Type FEM for Second Order Elliptic Problems[J]. Applied Mathematics and Mechanics, 2012, 33(2): 240-249. doi: 10.3879/j.issn.1000-0887.2012.02.009
[9]	LI Kai-tai, SHI Feng. Hodograph Method of Flow on Two-Dimensional Manifold[J]. Applied Mathematics and Mechanics, 2010, 31(3): 337-350. doi: 10.3879/j.issn.1000-0887.2010.03.009
[10]	GUO Yan, LIU Ru-xun. Characteristic-Based Finite Volume Scheme for 1D Euler Equations[J]. Applied Mathematics and Mechanics, 2009, 30(3): 291-300.
[11]	XU Xin-sheng, WANG Ga-ping, SUN Fa-ming. Analytical and Numerical Method of Symplectic System for Stokes Flow in the Two-Dimensional Rectangular Domain[J]. Applied Mathematics and Mechanics, 2008, 29(6): 639-648.
[12]	LIU Yang, LI Hong, HE Siriguleng. Mixed Time Discontinuous Space-Time Finite Element Method for Convection Diffusion Equations[J]. Applied Mathematics and Mechanics, 2008, 29(12): 1435-1442.
[13]	ZHOU Zhen-gong, WANG Biao. Investigation of the Behavior of a Model-Ⅰ Interface Crack in Piezoelectric Materials by Using the Schmidt Method[J]. Applied Mathematics and Mechanics, 2006, 27(7): 765-774.
[14]	QIN Xin-qiang, MA Yi-chen, ZHANG Yin. Two-Grid Method for Characteristics Finite-Element Solution of 2D Nonlinear Convection-Dominated Diffusion Problem[J]. Applied Mathematics and Mechanics, 2005, 26(11): 1365-1372.
[15]	GAO Zhi, DAI Min-guo, LI Gui-bo, BAI Wei. Perturbational Finite Volume Method for the Solution of 2-D Navier-Stokes Equations on Unstructured and Structured Colocated Meshes[J]. Applied Mathematics and Mechanics, 2005, 26(2): 222-230.
[16]	HUANG Zheng-hong. On Cauchy Problems for the RLW Equation in Two Space Dimensions[J]. Applied Mathematics and Mechanics, 2002, 23(2): 157-164.
[17]	Mei Liquan. A Spectral Streamline Diffusion Finite Element Coupled Method of Unsteady Transport Equation in the Field of Neutron Logging[J]. Applied Mathematics and Mechanics, 1999, 20(7): 691-698.
[18]	She Yinghe, Sun Ying, Guo Xiaoming. Free Boundary Problem of the 2D Seepage Flow[J]. Applied Mathematics and Mechanics, 1996, 17(6): 523-527.
[20]	Huang Si-xun. On Discontinuous Period Solution and Discontinuous Solitary Wave of Two-Dimension Shallow Water Equation[J]. Applied Mathematics and Mechanics, 1987, 8(9): 805-812.