### EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS

Xuhong Yu, Lusha Jin, Zhongqing Wang

1. School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
• Received:2018-12-20 Revised:2019-06-03 Online:2021-01-15 Published:2021-03-11
• Supported by:
This work was supported by Natural Science Foundation of China (Nos. 11571238, 11601332 and 11871043).

Xuhong Yu, Lusha Jin, Zhongqing Wang. EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(1): 43-62.

Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation, third-order equation, third-order KdV equation and fifth-order Kawahara equation 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 are expanded as an infinite and truncated Fourier-like series, respectively. Numerical experiments illustrate the effectiveness of the suggested approaches.

CLC Number:

 [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.[2] C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, Vol. V, P. G. Ciarlet and J. L. Lions, eds., Techniques of Scientific Computing (Part 2), Elsevier, Amsterdam, 1997.[3] J.L. Bona, S.M Sun and B.Y Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28(2003), 1391-1436.[4] J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn., Dover, New York, 2001.[5] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer, Berlin, 2006.[6] T. Colin and J.M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Diff. Eq., 6(2001), 1463-1492.[7] L. Fernández, F. Marcellán, T.E. Pérez, Miguel A. Piñar and Yuan Xu, Sobolev orthogonal polynomials on product domains, J. Comput. Anal. Appl., 284(2015), 202-215.[8] D. Funaro, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin, 1992.[9] O. Goubet and J. Shen, On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval, Adv. Differential Equations, 12(2007), 221-239.[10] B.Y. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.[11] B.Y. Guo, S.N. He and H.P. Ma, Chebyshev spectral-finite element method for two dimensional unsteady Navier-Stokes equation, J. Comput. Math., 20(2002), 65-78.[12] B.Y. Guo and J. Li, Fourier-Chebyshev spectral method for the two-dimensional Navier-Stokes equations, SIAM J. Numer. Anal., 33(1996), 1169-1187.[13] B.Y. Guo, H.P. Ma, W.M. Cao and H. Huang, The Fourier-Chebyshev spectral method for solving two-dimensional unsteady vorticity equations, J. Comput. Phys., 101(1992), 207-217.[14] W.Z. Huang and D.M. Sloan, The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal., 29(1992), 1626-1647.[15] R. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Jpn., 33(1972), 260-264.[16] J.M. Li, Z.Q. Wang and H.Y. Li, Fully diagonalized Chebyshev spectral methods for second and fourth order elliptic boundary value problems, Inter. J. Numer. Anal. Modeling, 15(2018), 243-259.[17] H.P. Ma and B.Y. Guo, The Chebyshev spectral method for Burgers-like equations, J. Comput. Math., 6(1988), 48-53.[18] 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.[19] 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.[20] F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math., 33(2015), 308-352.[21] W.J. Merryfield and B. Shizgal, Properties of collocation third-derivative operators, J. Comput. Phys., 105(1993), 182-185.[22] 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.[23] 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.[24] J. Shen and T. Tang, Spectral and High-order Methods with Applications, Science Press, Beijing, 2006.[25] J. Shen, T. Tang and L.L. Wang, Spectral methods:Algorithms, Analysis and Applications, Springer-Verlag, Berlin, 2011.[26] 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.[27] 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.[28] J.M. Yuan, J. Shen and J.H. Wu, A dual-Petrov-Galerkin method for the Kawahara-type equations, J. Sci. Comput., 34(2008), 48-63.
 [1] Changfeng Ma. A FEASIBLE SEMISMOOTH GAUSS-NEWTON METHOD FOR SOLVING A CLASS OF SLCPS [J]. Journal of Computational Mathematics, 2012, 30(2): 197-222. [2] Li Ping HE,Shun Kai SUN. THE PREDICTION-CORRECTION LEGENDRE COLLOCATION METHOD FOR NONLINEAR EVOLUTIONARY PROBLEMS [J]. Journal of Computational Mathematics, 2004, 22(5): 753-768. [3] I.D. Coope C.J. Price. A DIRECT SEARCH FRAME-BASED CONJUGATE GRADIENTS METHOD [J]. Journal of Computational Mathematics, 2004, 22(4): 489-500.
Viewed
Full text

Abstract