Solving unsteady Schrödinger equation using the improved element-free Galerkin method

doi:10.1088/1674-1056/25/2/020203

中国物理B ›› 2016, Vol. 25 ›› Issue (2): 20203-020203.doi: 10.1088/1674-1056/25/2/020203

Solving unsteady Schrödinger equation using the improved element-free Galerkin method

Rong-Jun Cheng(程荣军) and Yu-Min Cheng(程玉民)

1. Faculty of Maritime and Transportation, Ningbo University, Ningbo 315211, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

收稿日期:2015-07-09 修回日期:2015-11-01 出版日期:2016-02-05 发布日期:2016-02-05
通讯作者: Yu-Min Cheng E-mail:ymcheng@shu.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 11171208), the Natural Science Foundation of Zhejiang Province, China (Grant No. LY15A020007), the Natural Science Foundation of Ningbo City (Grant No. 2014A610028), and the K. C. Wong Magna Fund in Ningbo University, China.

Solving unsteady Schrödinger equation using the improved element-free Galerkin method

Rong-Jun Cheng(程荣军)¹ and Yu-Min Cheng(程玉民)²

1. Faculty of Maritime and Transportation, Ningbo University, Ningbo 315211, China;
2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

Received:2015-07-09 Revised:2015-11-01 Online:2016-02-05 Published:2016-02-05
Contact: Yu-Min Cheng E-mail:ymcheng@shu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 11171208), the Natural Science Foundation of Zhejiang Province, China (Grant No. LY15A020007), the Natural Science Foundation of Ningbo City (Grant No. 2014A610028), and the K. C. Wong Magna Fund in Ningbo University, China.

摘要/Abstract

摘要：

By employing the improved moving least-square (IMLS) approximation, the improved element-free Galerkin (IEFG) method is presented for the unsteady Schrödinger equation. In the IEFG method, the two-dimensional (2D) trial function is approximated by the IMLS approximation, the variation method is used to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. Because the number of coefficients in the IMLS approximation is less than in the moving least-square (MLS) approximation, fewer nodes are needed in the entire domain when the IMLS approximation is used than when the MLS approximation is adopted. Then the IEFG method has high computational efficiency and accuracy. Several numerical examples are given to verify the accuracy and efficiency of the IEFG method in this paper.

关键词: meshless method, improved moving least-square (IMLS) approximation, improved element-free Galerkin (IEFG) method, Schrö, dinger equation

Abstract:

Key words: meshless method, improved moving least-square (IMLS) approximation, improved element-free Galerkin (IEFG) method, Schrö, dinger equation

中图分类号: (Ordinary and partial differential equations; boundary value problems)

02.60.Lj

03.65.Ge (Solutions of wave equations: bound states)

程荣军, 程玉民. Solving unsteady Schrödinger equation using the improved element-free Galerkin method[J]. 中国物理B, 2016, 25(2): 20203-020203.

Rong-Jun Cheng(程荣军) and Yu-Min Cheng(程玉民). Solving unsteady Schrödinger equation using the improved element-free Galerkin method[J]. Chin. Phys. B, 2016, 25(2): 20203-020203.

参考文献 56

[1]	Kivshar Y and Agrawal G P 2003 Optical Solitons : From Fibers to Photonic Crystals (New York: Academic Press)
[2]	Deng Y X, Tu C H and Lu F Y 2009 Acta Phys. Sin. 58 3173 (in Chinses)
[3]	Zhao L, Sui Z, Zhu Q H, Zhang Y and Zuo Y L 2009 Acta Phys. Sin. 58 4731 (in Chinses)
[4]	Hasegawa A 1989 Optical Solitons in Fibers (Berlin: Springer-Verlag)
[5]	Dodd R K, Eilbeck J C, Gibbon J D and Morris H C 1982 Solitons and Nonlinear Wave Equations (New York: Academic Press)
[6]	Pitaevskii L P and Stringari S 2003 Bose-Einstein Condensation (Oxford: Oxford University Press)
[7]	Scott A 1999 Nonlinear Science: Emergence and Dynamics of Coherent Structures (Vol. 1) (Oxford: Oxford University Press)
[8]	Arnold A 1998 VLSI Design 6 313
[9]	Levy M 2000 Institution of Electrical Engineers, London
[10]	Tappert F D 1977 Lecture Notes in Physics (Berlin: Springer)
[11]	Kopylov Y V, Popov A V and Vinogradov A V 1995 Opt. Commun. 118 619
[12]	Huang W, Xu C, Chu S T and Chaudhuri S K 1992 J. Lightwave Technology 10 295
[13]	Subasi M 2002 Numerical Methods for Partial Differential Equations 18 752
[14]	Antoine X, Besse C and Mouysett V 2004 Math. Comput. 73 1779
[15]	Kalita J C, Chhabra P and Kumar S 2006 J. Comput. Appl. Math. 197 141
[16]	Dehghan M and Shokri A 2007 Comput. Math. Appl. 54 136
[17]	Dehghan M 2006 Math. Comput. Simul. 71 16
[18]	Dehghan M and Mirzaei D 2008 Engineering Analysis with Boundary Elements 32 747
[19]	Ang W T and Ang K C 2004 Numerical Methods for Partial Differen-tial Equations 20 843
[20]	Belytschko T, Krongauz Y, Organ D, Fleming M and Krysl P 1996 Comput. Method Appl. Mech. Eng. 139 3
[21]	Cheng Y M and Li J H 2006 Sci. China Ser. G Phys. Mech. Astron. 49 46
[22]	Cheng R J and Cheng Y M 2008 Appl. Numer. Math. 58 884
[23]	Peng M J and Cheng Y M 2009 Engineering Analysis with Boundary Elements 33 77
[24]	Li D M and Bai F N and Cheng Y M and Liew K M 2012 Comput. Methods Appl. Mech. Eng. 233-236 1
[25]	Lancaster P and Salkauskas K 1981 Math. Comput. 37 141
[26]	Cheng Y M and Chen M J 2003 Acta Mech. Sin. 35 181
[27]	Cheng Y M and Peng M J 2005 Sci. China-Ser. G Phys. Mech. Astron. 48 641
[28]	Liew K M, Cheng Y M and Kitipornchai S 2006 Int. J. Numer. Methods Eng. 65 1310
[29]	Kitipornchai S, Liew K M and Cheng Y M 2005 Comput. Mech. 36 13
[30]	Liew K M, Cheng Y M and Kitipornchai S 2005 Int. J. Numer. Methods Eng. 64 1610
[31]	Zhang Z, Liew K M and Cheng Y M 2008 Engineering Analysis with Boundary Elements 32 100
[32]	Zhang Z, Liew K M, Cheng Y M and Li Y Y 2008 Engineering Anal-ysis with Boundary Elements 32 241
[33]	Zhang Z, Wang J F, Cheng Y M and Liew K M 2013 Sci. China-Phys. Mech. Astron. 56 1568
[34]	Zhang Z, Cheng Y M and Liew K M 2013 Engineering Analysis withBoundary Elements 37 1576
[35]	Ren H P, Cheng Y M and Zhang W 2009 Chin. Phys. B 18 4065
[36]	Ren H P, Cheng Y M and Zhang W 2010 Sci. China-Ser. G Phys. Mech. Astron. 53 758
[37]	Ren H P and Cheng Y M 2011 Int. J. Appl. Mech. 3 735
[38]	Ren H P and Cheng Y M 2012 Engineering Analysis with Boundary Elements 36 873
[39]	Sun F X, Wang J F, Cheng Y M and Huang A X 2015 Appl. Numer. Math. 98 79
[40]	Cheng R J and Liew K M 2012 Engineering Analysis with Boundary Elements 36 203
[41]	Cheng R J and Liew K M 2009 Comput. Mech. 45 1
[42]	Cheng R J and Liew K M 2012 Engineering Analysis with Boundary Elements 36 1322
[43]	Cheng R J and Liew K M 2012 Comput. Method Appl. Mech. Eng. 245-246 132
[44]	Wang J F, Sun F X and Cheng Y M 2012 Chin. Phys. B 21 090204
[45]	Wang J F, Wang J F, Sun F X and Cheng Y M 2013 Int. J. Comput. Methods 10 1350043
[46]	Sun F X, Wang J F, Cheng Y M and Huang A X 2015 Appl. Numer. Math. 98 79
[47]	Chen L and Cheng Y M 2010 Sci. China-Ser. G Phys. Mech. Astron. 53 954
[48]	Chen L and Cheng Y M 2010 Chin. Phys. B 19 090204
[49]	Chen L, Cheng Y M and Ma H P 2015 Comput. Mech. 55 591
[50]	Peng M J, Liu P and Cheng Y M 2009 Int. J. Appl. Mech. 1 367
[51]	Peng M J, Li D M and Cheng Y M 2011 Engineering Structure 33 127
[52]	Li D M, Peng M J and Cheng Y M 2011 Sci. China-Ser. G Phys. Mech. Astron. 41 1003
[53]	Cheng Y M, Li R X and Peng M J 2012 Chin. Phys. B 21 090205
[54]	Bai F N, Li D M, Wang J F and Cheng Y M 2012 Chin. Phys. B 21 020204
[55]	Cheng Y M, Liu C, Bai F N and Peng M J 2015 Chin. Phys. B 24 100202
[56]	Deng Y J, Liu C, Peng M J and Cheng Y M 2015 Int. J. Appl. Mech. 7 1550017

Solving unsteady Schrödinger equation using the improved element-free Galerkin method

Solving unsteady Schrödinger equation using the improved element-free Galerkin method

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