Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations

Shan Li; Shi-Mi Yan; Zhong-Qing Wang

doi:10.3934/dcdsb.2019239

Article Contents

2020, Volume 25, Issue 4: 1543-1563. Doi: 10.3934/dcdsb.2019239

This issue Previous Article Critical and super-critical abstract parabolic equations Next Article Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations

School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

^* Corresponding author
^* Corresponding author

Received: January 2019

Revised: June 2019

Early access: November 2019

Published: April 2020

This work was supported in part by National Natural Science Foundation of China (Nos. 11601331 and 11571238)

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Abstract

Efficient Legendre dual-Petrov-Galerkin methods for solving odd-order differential equations are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like series. Numerical results indicate that the suggested methods are extremely accurate and efficient, and suitable for the odd-order equations.

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Mathematics Subject Classification: 65N35, 33C45, 35J58.

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Figure 1. Numerical errors of scheme (20)

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Figure 2. Numerical errors of scheme (20)

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Figure 3. Numerical errors of scheme (41)

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Figure 4. Numerical errors of scheme (41)

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Figure 5. Numerical errors of scheme (59)

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Figure 6. Numerical errors of scheme (59)

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Table 1. Condition numbers of the usual Legendre dual-Petrov-Galerkin method for problem (19)

$ \rm Matrices $	$ N=20 $	$ N=40 $	$ N=60 $	$ N=80 $	$ N=100 $
$ \alpha(p_k, q_l) $	3.1644e+05	2.3889e+07	3.3847e+08	2.3036e+09	1.0362e+10
$ \beta(p_k, \partial_xq_l) $	1.5106e+04	3.6487e+05	2.5120e+06	1.0070e+07	2.9818e+07
$ \gamma(\partial_xp_k, \partial_xq_l) $	2.4015e+03	3.9365e+04	2.0296e+05	6.4966e+05	1.6010e+06
$ (\partial_xp_k, \partial^2_xq_l) $	2.5926e+02	2.3471e+03	8.4079e+03	2.0652e+04	4.1318e+04
$ \mathcal{A}(p_k, q_l) $	2.5961e+02	2.3422e+03	8.3910e+03	2.0616e+04	4.1257e+04

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Table 2. Condition numbers of our new Legendre dual-Petrov-Galerkin method for problem (19)

$ \rm Matrices $	$ N=20 $	$ N=40 $	$ N=60 $	$ N=80 $	$ N=100 $
$ \alpha(\varphi_k, \psi_l) $	8.2281e+02	1.0014e+04	4.6167e+04	1.3919e+05	3.3027e+05
$ \beta(\varphi_k, \partial_x\psi_l) $	4.2539e+01	1.6248e+02	3.6050e+02	6.3660e+02	9.9077e+02
$ \gamma(\partial_x\varphi_k, \partial_x\psi_l) $	1.3709e+00	1.3731e+00	1.3734e+00	1.3736e+00	1.3736e+00
$ (\partial_x\varphi_k, \partial^2_x\psi_l) $	1.3826e+00	1.3842e+00	1.3846e+00	1.3848e+00	1.3848e+00
$ \mathcal{A}(\varphi_k, \psi_l) $	1.0000e+00	1.0000e+00	1.0000e+00	1.0000e+00	1.0000e+00

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Table 3. Condition numbers of the usual Legendre dual-Petrov-Galerkin method for problem (39)

$ \rm Matrices $	$ N=20 $	$ N=40 $	$ N=60 $	$ N=80 $	$ N=100 $
$ \alpha(r_k, s_l) $	2.2919e+07	1.4122e+10	7.8699e+11	1.4829e+13	1.5013e+14
$ \beta(\partial_xr_k, \partial^2_xs_l) $	1.0527e+05	1.1264e+07	1.8445e+08	1.3612e+09	6.4521e+09
$ (\partial^2_xr_k, \partial^3_xs_l) $	3.7507e+03	1.3987e+05	1.1653e+06	5.2074e+06	1.6553e+07
$ \mathcal{B}(r_k, s_l) $	3.7593e+03	1.3996e+05	1.1656e+06	5.2083e+06	1.6555e+07

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Table 4. Condition numbers of our new Legendre dual-Petrov-Galerkin method for problem (39)

$ \rm Matrices $	$ N=20 $	$ N=40 $	$ N=60 $	$ N=80 $	$ N=100 $
$ \alpha(\Phi_k, \Psi_l) $	6.1180e+06	2.2260e+09	8.8437e+10	1.2957e+12	1.0730e+13
$ \beta(\partial_x\Phi_k, \partial^2_x\Psi_l) $	5.1022e+02	5.8057e+03	2.6067e+04	7.7483e+04	1.8226e+05
$ (\partial^2_x\Phi_k, \partial^3_x\Psi_l) $	1.0683e+00	1.0684e+00	1.0684e+00	1.0684e+00	1.0684e+00
$ \mathcal{B}(\Phi_k, \Psi_l) $	1.0000e+00	1.0000e+00	1.0000e+00	1.0000e+00	1.0000e+00

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References

[1]	Q. Ai, H.-Y. Li and Z.-Q. Wang, Diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for elliptic boundary value problems, Appl. Numer. Math., 127 (2018), 196-210. doi: 10.1016/j.apnum.2018.01.003.
[2]	C. Bernardi and Y. Maday, Spectral methods, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 5 (1997), 209-485. doi: 10.1016/S1570-8659(97)80003-8.
[3]	J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, NY, 2001.
[4]	C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-540-30726-6.
[5]	D. Funaro, Polynomial Approximation of Differential Equations, Lecture Notes in Physics. New Series m: Monographs, 8. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-540-46783-0.
[6]	D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. doi: 10.1137/1.9781611970425.
[7]	B.-Y. Guo, Spectral Methods and Their Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/3662.
[8]	W. Z. Huang and D. M. Sloan, The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal., 29 (1992), 1626-1647. doi: 10.1137/0729094.
[9]	J.-M. Li, Z.-Q. Wang and H.-Y. Li, Fully diagonalized Chebyshev spectral methods for second and fourth order elliptic boundary value problems, Int. J. Numer. Anal. Model., 15 (2018), 243-259. doi: 10.1016/j.apnum.2018.01.003.
[10]	F.-J. Liu, H.-Y. Li and Z.-Q. Wang, A fully diagonalized spectral method using generalized Laguerre functions on the half line, Adv. Comput. Math., 43 (2017), 1227-1259. doi: 10.1007/s10444-017-9522-3.
[11]	F.-J. Liu, H.-Y. Li and Z.-Q. Wang, Spectral methods using generalized Laguerre functions for second and fourth order problems, Numer. Algor., 75 (2017), 1005-1040. doi: 10.1007/s11075-016-0228-2.
[12]	H. P. Ma and W. W. Sun, A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations, SIAM J. Numer. Anal., 38 (2000), 1425-1438. doi: 10.1137/S0036142999361505.
[13]	H. P. Ma and W. W. Sun, Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation, SIAM J. Numer. Anal., 39 (2001), 1380-1394. doi: 10.1137/S0036142900378327.
[14]	W. J. Merryfield and B. Shizgal, Properties of collocation third-derivative operators, J. Comput. Phys., 105 (1993), 182-185. doi: 10.1006/jcph.1993.1065.
[15]	J. Shen, Efficient spectral-Galerkin method. Ⅰ. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505. doi: 10.1137/0915089.
[16]	J. Shen, Efficient spectral-Galerkin method. Ⅱ. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), 74-87. doi: 10.1137/0916006.
[17]	J. Shen, A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KdV equation, SIAM J. Numer. Anal., 41 (2003), 1595-1619. doi: 10.1137/S0036142902410271.
[18]	J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-71041-7.
[19]	J. Shen and L.-L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710-720. doi: 10.1016/j.apnum.2006.07.012.
[20]	J. Shen and L.-L. Wang, Legendre and Chebyshev dual-Petrov-Galerkin methods for hyperbolic equations, Comput. Methods Appl. Mech. Engrg., 196 (2007), 3785-3797. doi: 10.1016/j.cma.2006.10.031.