April  2020, 25(4): 1543-1563. doi: 10.3934/dcdsb.2019239

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

Received  January 2019 Revised  June 2019 Published  April 2020 Early access  November 2019

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

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.

Citation: Shan Li, Shi-Mi Yan, Zhong-Qing Wang. Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1543-1563. doi: 10.3934/dcdsb.2019239
References:
[1]

Q. AiH.-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. LiZ.-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. LiuH.-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. LiuH.-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.

show all references

References:
[1]

Q. AiH.-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. LiZ.-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. LiuH.-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. LiuH.-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.

Figure 1.  Numerical errors of scheme (20)
Figure 2.  Numerical errors of scheme (20)
Figure 3.  Numerical errors of scheme (41)
Figure 4.  Numerical errors of scheme (41)
Figure 5.  Numerical errors of scheme (59)
Figure 6.  Numerical errors of scheme (59)
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
$ \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
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
$ \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
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
$ \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
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
$ \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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