January  2014, 19(1): 299-322. doi: 10.3934/dcdsb.2014.19.299

Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations

1. 

Department of Mathematics, Shanghai Normal University, Division of Computational Science of E-institute of Shanghai Universities, Shanghai, 200234, China, China

Received  August 2012 Revised  September 2013 Published  December 2013

We propose an efficient Legendre-Gauss collocation algorithm for second-order nonlinear ordinary differential equations (ODEs). We also design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partial differential equations (PDEs), which can be implemented in a synchronous parallel fashion. Numerical results indicate the high accuracy and effectiveness of the suggested algorithms.
Citation: Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
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2nd edition, Springer, Berlin, Heidelberg, 2006.  Google Scholar

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L. J. Yi and Z. Q. Wang, Legendre-Gauss-type collocation algorithms for nonlinear ordinary/ partial differential equations,, Int. J. Comput. Math.., ().  doi: 10.1080/00207160.2013.841901.  Google Scholar

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show all references

References:
[1]

Numer. Methods Partial Differential Equations, 5 (1989), 363-399. doi: 10.1002/num.1690050407.  Google Scholar

[2]

Numer. Methods Partial Differential Equations, 6 (1990), 343-369. doi: 10.1002/num.1690060406.  Google Scholar

[3]

J. Comput. Phys., 119 (1995), 62-74. doi: 10.1006/jcph.1995.1116.  Google Scholar

[4]

(eds. P. G. Ciarlet and J. L. Lions), North-Holland, Amsterdam, 1997. doi: 10.1016/S1570-8659(97)80003-8.  Google Scholar

[5]

2nd edition, Dover Publications, New York, 2001.  Google Scholar

[6]

John Wiley & Sons, Chichester, 1987.  Google Scholar

[7]

Springer-Verlag, Berlin, 2006.  Google Scholar

[8]

Springer-Verlag, Berlin, 2007.  Google Scholar

[9]

Comput. Phys. Comm., 147 (2002), 770-787. doi: 10.1016/S0010-4655(02)00460-5.  Google Scholar

[10]

Numer. Algor., 26 (2001), 347-363. doi: 10.1023/A:1016629706668.  Google Scholar

[11]

Springer-Verlag, Berlin, 1992.  Google Scholar

[12]

J. Comput. Phys., 102 (1992), 88-97. doi: 10.1016/S0021-9991(05)80008-7.  Google Scholar

[13]

Philadelphia, Pa., 1977.  Google Scholar

[14]

World Scientific, Singapore, 1998. doi: 10.1142/3662.  Google Scholar

[15]

Adv. Comput. Math., 30 (2009), 249-280. doi: 10.1007/s10444-008-9067-6.  Google Scholar

[16]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1029-1054. Google Scholar

[17]

Appl. Numer. Math., 59 (2009), 1386-1408. doi: 10.1016/j.apnum.2008.08.007.  Google Scholar

[18]

Springer-Verlag, Berlin, 1987.  Google Scholar

[19]

Springer-Verlag, Berlin, 1991.  Google Scholar

[20]

2nd edition, Springer, Berlin, Heidelberg, 2006.  Google Scholar

[21]

Int. J. Numer. Anal. Model., 8 (2011), 86-104.  Google Scholar

[22]

SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944.  Google Scholar

[23]

John Wiley & Sons, Chichester, 1991.  Google Scholar

[24]

Phys. Lett. A, 289 (2001), 69-74. doi: 10.1016/S0375-9601(01)00580-1.  Google Scholar

[25]

SIAM J. Numer. Anal., 38 (2000), 837-875. doi: 10.1137/S0036142999352394.  Google Scholar

[26]

Calcolo, 37 (2000), 207-232. doi: 10.1007/s100920070002.  Google Scholar

[27]

Science Press, Beijing, 2006.  Google Scholar

[28]

Springer Series in Computational Mathematics, Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[29]

Appl. Numer. Math., 57 (2007), 710-720. doi: 10.1016/j.apnum.2006.07.012.  Google Scholar

[30]

Cambridge University Press, Cambridge, 1996.  Google Scholar

[31]

SIAM J. Numer. Anal., 23 (1986), 11-26. doi: 10.1137/0723002.  Google Scholar

[32]

SIAM J. Numer. Anal., 26 (1989), 1-11. doi: 10.1137/0726001.  Google Scholar

[33]

Adv. Comput. Math., 17 (2002), 349-367. doi: 10.1023/A:1016273820035.  Google Scholar

[34]

Appl. Numer. Math., 57 (2007), 1-11. doi: 10.1016/j.apnum.2005.11.009.  Google Scholar

[35]

J. Sci. Comput., 52 (2012), 226-255. doi: 10.1007/s10915-011-9538-7.  Google Scholar

[36]

J. Sci. Comput., 25 (2005), 523-549. doi: 10.1007/s10915-004-4796-2.  Google Scholar

[37]

BIT Numer. Math., 52 (2012), 773-795. doi: 10.1007/s10543-012-0379-z.  Google Scholar

[38]

L. J. Yi and Z. Q. Wang, Legendre-Gauss-type collocation algorithms for nonlinear ordinary/ partial differential equations,, Int. J. Comput. Math.., ().  doi: 10.1080/00207160.2013.841901.  Google Scholar

[39]

Commun. Appl. Math. Comput., 26 (2012), 223-238.  Google Scholar

[40]

Comput. Methods Appl. Mech. Engrg., 116 (1994), 135-146. doi: 10.1016/S0045-7825(94)80017-0.  Google Scholar

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