二维抛物型问题的Strang型交替分段区域分裂格式

计算物理 ›› 2018, Vol. 35 ›› Issue (4): 413-428.

二维抛物型问题的Strang型交替分段区域分裂格式

张守慧¹, 梁栋²

1. 济南大学数学科学学院, 济南 250022;
2. 约克大学, 多伦多, 加拿大 M3J1P3

收稿日期:2017-05-02 修回日期:2017-08-16 出版日期:2018-07-25 发布日期:2018-07-25
作者简介:ZHANG Shouhui(1978-),female, Doctor, lecturer, major in numerical methods for partial difference equations,E-mail:ss_zhangsh@ujn.edu.cn
基金资助:
Supported by the Doctoral Fund of Shandong(BS2013NJ016) and sponsored by SRF for ROCS, SEM

A Strang-type Alternating Segment Domain Decomposition Method for Two-dimensional Parabolic Equations

ZHANG Shouhui¹, LIANG Dong²

1. School of Mathematical Sciences, University of Jinan, Jinan 250022, China;
2. Department of Mathematics and Statistics, York University, Toronto, ON, M3J1P3 Canada

Received:2017-05-02 Revised:2017-08-16 Online:2018-07-25 Published:2018-07-25
Supported by:
Supported by the Doctoral Fund of Shandong(BS2013NJ016) and sponsored by SRF for ROCS, SEM

摘要/Abstract

摘要： 给出求解二维抛物型方程的Strang型的交替分段区域分裂格式。交替分段思想可以将区域分为一些不重叠的子区域，Strang型算子分裂技巧通过将高维问题的求解分解为几个低维问题的求解来降低其求解的复杂度。方法是无条件稳定的，理论分析了截断误差。数值算例说明格式的有效性及时空的二阶精度.

关键词: Strang型, 交替分段格式, 区域分裂, 抛物型问题, 并行算法

Abstract: A Strang-type alternating segment domain decomposition method for 2-D parabolic problems is proposed. The domain can be divided into non-overlapping multi-block sub-domains by the idea of alternating segments. Strang-type splitting technique reduces complexity of the solving of the high dimensional problems by a series of one-dimensional ones. The method is proved to be unconditionally stable and truncation error is analyzed. Numerical experiments show that convergence rates in time and space are both second order.

Key words: Strang-type, alternating segment scheme, domain decomposition, parabolic problem, parallel algorithm

中图分类号:

O241.82

张守慧, 梁栋. 二维抛物型问题的Strang型交替分段区域分裂格式[J]. 计算物理, 2018, 35(4): 413-428.

ZHANG Shouhui, LIANG Dong. A Strang-type Alternating Segment Domain Decomposition Method for Two-dimensional Parabolic Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(4): 413-428.

参考文献

[1] CHORIN A J, MARSDON J E. A mathematical introduction to fluid mechanics[M]. 2nd ed. Springer-Verlag, 1990.
[2] AZIZ K, SETTARI A. Petrolem reservoir simulation[M]. London:Applied Science Publisher, Ltd, 1979.
[3] BEAR J. Hydraulics of groundwater[M]. New York:McGraw-Hill, 1978.
[4] EWING R E. The mathematics of reservoir simulation[J].Frontiers in Applied Mathematics, 1988,1:294-318.
[5] LV Guixia, SUN Shunkai. A finite directional difference meshless method for diffusion equations[J]. Chinese J Comput Phys, 2015, 32(6):649-661.
[6] EVANS D J, ABDULLAH A R B. Group explicit methods for parabolic equations[J]. Inter J Comp Math, 1983, 14:73-105.
[7] EVANS D J, ABDULLAH A R B. A new explicit method for the diffusion-convection equation[J]. Comput Math Appl, 1985,11:145-154.
[8] ZHANG B, YUAN G, et al. The parallel finite difference methods for partial differential equation[M]. Beijing:Science Press Beijing, 1994:203-218.
[9] DOUGLAS Jr J, PEACEMAN D. Numerical solution of two-dimensional heat flow problems[J]. American Institute of Chemical Engineering Journal, 1955, 1:505-512.
[10] PEACEMAN D, RACHEFORD H. The numerical solution of parabolic and elliptic equations[J]. J Soc Indus Appl Math, 1955,3:28-41.
[11] DOUGLAS J Jr, KIM S. Improved accuracy for locally one-dimensional methods for parabolic equations[J]. Math Models Appl Sci, 2001, 11:1563-1579.
[12] D'YAKONOV E. Difference schemes with split operators for multidimensional unsteady problems(English translation)[J]. USSR Comp Math, 1963, 3:581-607.
[13] MARCHUK G. Methods of numerical mathematics[M]. New York, Heidelberg, and Berlin:Springer-Verlag, 1982.
[14] MARCHUK G I. Splitting and alternating direction methods[M]//Ciarlet P G, Lions J L, eds.Handbook of numerical analysis, Vol. I. Finite difference methods(Part I), Amsterdam:Elsevier Science Publishers, 1990, 197-462.
[15] YANENKO N N. The method of fractional steps[M]. Berlin:Springer-Verlag, 1971.
[16] STRANG G. Accurate partial difference methods I:Linear Cauchy problems[J]. Arch Rational Mech Anal, 1963, 12:392-402.
[17] STRANG G. On the construction and comparison of difference schemes[J]. SIAM J Numer Anal, 1968, 3:506-517.
[18] DU Q, MU M, WU Z N. Efficient parallel algorithms for parabolic problems[J]. SIAM J Numer Anal, 2001, 39:1469-1487.
[19] LIANG D, DU C, WANG H. A fractional step ELLAM approach to high dimension convection diffusion problems with forward particle tracking[J]. J Comput Phys, 2007, 221:198-225.
[20] SMITH B F, BJOST P E, GROPP W D. Domain decomposition methods for partial differential equations[M]. Cambridge:Cambridge University Press, 1996.
[21] QIN J. The new alternating direction implicit difference methods for solving three-dimensional parabolic equations[J]. Applied Mathematics Modelling, 2010, 34:890-897.
[22] LEVEQUE R J, OLIGER J. Numerical methods based on additive splitting for hyperbolic partial differencial equation[J]. Math Comput, 1983, 40:469-497.
[23] SPORTISSE B,BENCTEUX G, PLION P. Method of lines versus operator splitting for reaction-diffusion systems with fast chemistry[M]. Enviromental Modelling Software, 2000, 15:673-679.
[24] DAOUD D S. On combination of first order Strang's splitting and additive splitting for the solution of multi-dimensional convection diffusion problem[J]. Inter J Comp Math, 2007, 84(12):1781-1794.
[25] KELLOGG R B. An alternating direction method for operator equations[J]. SIAM, 1964, 12:848-854.
[26] DCMKOWICZ L, ODEN T J, RACHOWICZ W. A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting mehods and h-p adaptivity[J]. Comput Methods Appl Mech Engrg, 1990, 84:275-326.
[27] LAPIDUS L, PINDER G F. Numerical solution of partial differntial equations in science and engineering[M]. NY:John Wiley and Sons, 1982.
[28] KHAN L A, LIU P L F. An operator splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations[J]. Comput Methods App Mech Engrg, 1995, 127:181-201.
[29] KHAN L A, LIU P LF. Numerical analyses of operator-splitting algorithms for the two-dimensional advection-diffusion equation[J]. Comput Methods Appl Mech Engrg, 1998, 152:337-359.
[30] CAO J, XU X, WANG Y. Parallel algorithms for separable elliptic equation based on GPU[J]. Chinese J Comput Phys, 2015, 32(4):475-481.