ZHAN Yong-qiang, ZHANG Chuan-lin. A Family of High Accuracy Implicit Difference Schemes for Solving Parabolic Equations[J]. Applied Mathematics and Mechanics, 2014, 35(7): 790-797. doi: 10.3879/j.issn.1000-0887.2014.07.008
 Citation: ZHAN Yong-qiang, ZHANG Chuan-lin. A Family of High Accuracy Implicit Difference Schemes for Solving Parabolic Equations[J]. Applied Mathematics and Mechanics, 2014, 35(7): 790-797.

# A Family of High Accuracy Implicit Difference Schemes for Solving Parabolic Equations

##### doi: 10.3879/j.issn.1000-0887.2014.07.008
Funds:  The National Natural Science Foundation of China(61070165)
• Rev Recd Date: 2014-05-15
• Publish Date: 2014-07-15
• A family of implicit difference schemes with high accuracy for solving 1-dimensional parabolic equations were given. First, a difference approximation expression of the first order partial derivative of the solution to the parabolic equation was deduced at the special nodes; then this difference approximation expression and the second order central difference quotient approximation were used to construct a family of implicit difference schemes by the method of undetermined coefficients, and appropriate parameters were chosen to endow the schemes with high order truncation errors. In turn, the new difference schemes were proved to be stable as long as r was more than 1/6 with the Fourier analysis method. Finally, a numerical experiment was conduted on comparison of accuracy between the exact solutions, results of the new difference schemes and those of the other schemes with the same order truncation errors, as well as comparison of computational efficiency between the new schemes and the classical implicit difference schemes. The results demonstrate the high accuracy and efficiency of the presented difference schemes.
•  [1] 陆金甫, 关治. 偏微分方程数值解法[M]. 北京: 清华大学出版社, 2010: 82-85.（LU Jin-fu, GUAN Zhi. Numerical Solution of Partial Differential Equations[M]. Beijing: Tsinghua University Press, 2010: 82-85.（in Chinese）） [2] 戴嘉尊, 邱建贤. 微分方程数值解法[M]. 南京: 东南大学出版社, 2008: 47-56, 85-87.（DAI Jia-zun, QIU Jian-xian. Numerical Solution of Differential Equations[M]. Nanjing: Southeast University Press, 2008: 47-56, 85-87.（in Chinese）） [3] GAO Jia-quan, HE Gui-xia. An unconditionally stable parallel difference scheme for parabolic equations[J]. Applied Mathematics and Computation,2003,135(2/3): 391-398. [4] Sapagovas M. On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems[J]. Lithuanian Mathematical Journal,2008,48(3): 339-356. [5] Cash J R. Two new finite difference schemes for parabolic equations[J]. SIAM Journal on Numerical Analysis,1982,21(3): 433-446. [6] Ekolin G. Finite difference methods for a nonlocal boundary value problem for the heat equation[J]. BIT Numerical Mathematics,1991,31(2): 245-261. [7] LIU Yun-kang. Numerical solution of the heat equation with nonlocal boundary conditions[J]. Journal of Computational and Applied Mathematics,1999,110(1): 115-127. [8] Gulin A, Ionkin N, Morozova V. Stability criterion of difference schemes for the heat conduction equation with nonlocal boundary conditions[J]. Computational Methods in Applied Mathematics,2006,6(1): 31-55. [9] SUN Ping, LUO Zhen-dong, ZHOU Yan-jie. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations[J]. Applied Numerical Mathematics,2010,60(1/2): 154-164. [10] Borovykh N. Stability in the numerical solution of the heat equation with nonlocal boundary conditions[J]. Applied Numerical Mathematics,2002,42(1/3): 17-27. [11] 马明书. 一维抛物型方程的一个新的高精度显示差分格式[J]. 数值计算与计算机应用, 2001(2): 156-160.（MA Ming-shu. A new high accuracy explicit difference scheme with branching stable for solving parabolic equation of one-dimension[J]. Journal on Numerical Methods and Computer Applications,2001(2): 156-160.（in Chinese）） [12] 马驷良. 二阶矩阵族Gn(k,Δt)一致有界的充要条件及其对差分方程稳定性的应用[J]. 高等学校计算数学学报, 1980,2(2): 41-53.（MA Si-liang. The necessary and sufficient condition for the two-order matrix family Gn(k,Δt) uniformly bounded and its applications to the stability of difference equations[J]. Numerical Mathematics: A Journal of Chinese Universities,1980,2(2): 41-53.（in Chinese））

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