Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T11:37:09.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixed Spectral Method for Heat Transfer Using Generalised Hermite Functions and Legendre Polynomials

Part of: Parabolic equations and systems Basic methods in fluid mechanics Partial differential equations, boundary value problems

Published online by Cambridge University Press:  19 October 2016

Tian-Jun Wang ,
Chao Zhang  and
Qiong Zhang
Tian-Jun Wang*
Affiliation:
Henan University of Science and Technology, Luoyang, 471003, China
Chao Zhang
Affiliation:
Jiangsu Normal University, Xuzhou, 221116, China Jiangsu Key Laboratory of Education Big Data Science and Engineering, Xuzhou, 221116, China
Qiong Zhang
Affiliation:
Henan University of Science and Technology, Luoyang, 471003, China
*
*Corresponding author. Email address:[email protected] (T.-J. Wang)
Get access

Abstract

We propose amixed spectral method for heat transfer in unbounded domains, using generalised Hermite functions and Legendre polynomials. Some basic results on the mixed generalised Hermite-Legendre orthogonal approximation are established, which plays important roles in spectral methods for various problems defined on unbounded domains. As an example, the mixed generalised Hermite-Legendre spectral scheme is constructed for anisotropic heat transfer. Its convergence is proven, and some numerical results demonstrate the spectral accuracy of this approach.

Keywords

The scaled generalised Hermite functionsmixed spectral methodanisotropic heat transferunbounded domains

MSC classification

Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 76M22: Spectral methods 65N12: Stability and convergence of numerical methods 35K15: Initial value problems for second-order parabolic equations 35K20: Initial-boundary value problems for second-order parabolic equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 448 - 465
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bernardi, C. and Maday, Y., Spectral methods, in Handbook of Numerical Analysis, Ciarlet, P. G. and Lions, J. L. (Eds.), pp. 209–486, Elsevier, Amsterdam (1997).Google Scholar
[2] Bergh, J. and Löfström, J., Interpolation Spaces, An Introduction, Springer, Berlin (1976).CrossRefGoogle Scholar
[3] Boyd, J.P., Chebyshev and Fourier Spectral Methods, 2nd edition, Dover, New York (2001).Google Scholar
[4] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods, Fundamentals in Single Domains, Springer, Berlin (2006).CrossRefGoogle Scholar
[5] Funaro, D., Polynomial Approximations of Differential Equations, Springer, Berlin (1992).CrossRefGoogle Scholar
[6] Funaro, D., A variational formulation for the Chebyshev pseudospectral approximation of Neumann problems, SIAM J. Numer. Anal. 27, 695–703 (1990).CrossRefGoogle Scholar
[7] Guo, B.Y., Spectral Methods and Their Applications, World Scientific, Singapore (1998).CrossRefGoogle Scholar
[8] Guo, B.Y., Shen, J. and Wang, L.L., Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comp. 27, 305–322 (2006).CrossRefGoogle Scholar
[9] Guo, B.Y. and Wang, L.L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128, 1–41 (2004).CrossRefGoogle Scholar
[10] Gottlieb, D. and Orszag, S.A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia (1977).CrossRefGoogle Scholar
[11] Coulaud, O., Funaro, D. and Kavian, O., Laguerre spectral approximation of elliptic problems in exterior domains, Comp. Mech. Appl. Mech. Eng. 80, 451–458 (1990).CrossRefGoogle Scholar
[12] Fok, J.C.M., Guo, B.Y. and Tang, T., Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comp. 71, 1497–1528 (2001).CrossRefGoogle Scholar
[13] Funaro, D. and Kavian, O., Approximation of some diffusion evolution equation in unbounded domains by Hermite function, Math. Comp. 57, 597–619 (1999).CrossRefGoogle Scholar
[14] Guo, B.Y., Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp. 68, 1069–1078 (1999).CrossRefGoogle Scholar
[15] Guo, B.Y., Wang, L.L. and Wang, Z.Q., Generalized Laguerre interpolation and pseudospectral method for unbounded domains, SIAM J. Numer. Anal. 43, 2567–2589 (2006).Google Scholar
[16] Guo, B.Y., Shen, J. and Xu, C.L., Spectral and pseudospectral approximation using Hermite functions: Application to the Dirac equation, Adv. Comp. Math. 19, 35–55 (2003).CrossRefGoogle Scholar
[17] Guo, B.Y., Shen, J. and Xu, C.L., Generalized Laguerre approximation and its applications to exterior problems, J. Comp. Math. 23, 113–130 (2005).Google Scholar
[18] Guo, B.Y., and Xu, C.L., Hermite pseudospectral method for nonlinear differential equations, RAIRO Math. Model. Numer. Anal. 34, 859–872 (2000).CrossRefGoogle Scholar
[19] Guo, B.Y. and Wang, T.J., Mixed Legendre-Hermite spectral method for heat transfer in an infinite plate, Comput. Math. Appl. 51, 751–768 (2006).CrossRefGoogle Scholar
[20] Guo, B.Y. and Wang, T.J., Composite Laguerre-Legendre spectral method for exterior problems, Adv. Comp. Math. 32, 393–429 (2010).CrossRefGoogle Scholar
[21] Szegö, G., Orthogonal Polynomials, 4th ed., Amer.Math. Soc. Colloq. Publ. 23, AMS, Providence (1975).Google Scholar
[22] Le Bourdiec, S., de Vuyst, F. and Jacquet, L., Numerical solution of the Vlasov-Poisson system using generalized Hermite functions, Comput. Phys. Comm. 175, 528–544 (2006).CrossRefGoogle Scholar
[23] Aguirre, J. and Rivas, J., A spectral viscosity method based on Hermite functions for nonlinear conservation laws, SIAM J. Numer. Anal. 46, 1060–1078 (2008).CrossRefGoogle Scholar
[24] Ma, H.P. and Zhao, T.G., A stabilized Hermite spectral method for second-order differential equations in unbounded domains, Numer. Meth. Partial Differential Equations 23, 968–983 (2007).CrossRefGoogle Scholar
[25] Ma, H.P., Sun, W.W. and Tang, T., Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains, SIAM J. Numer. Anal. 43, 58–75 (2005).CrossRefGoogle Scholar
[26] Shen, J., Tang, T. and Wang, L.L., Spectral Method: Algorithms, Analysis and Applications, Springer, Berlin (2011).CrossRefGoogle Scholar
[27] Tang, T., The Hermite spectral method for Gauss-type function, SIAM J. Sci. Comput. 14, 594–606 (1993).CrossRefGoogle Scholar
[28] Wang, Z.Q., Guo, B.Y. and Zhang, W., Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions, J. Comput. Appl. Math. 217, 271–298 (2008).CrossRefGoogle Scholar
[29] Wang, T.J., Generalized Laguerre spectral method for Fisher's equation on a semi-infinite interval, Inter. J. Comput. Math. 92, 1039–1052 (2015).CrossRefGoogle Scholar
[30] Wang, T.J., Mixed spectral method for heat transfer with inhomogeneous Neumann boundary condition in an infinite strip, Appl. Numer. Math. 92, 82–97 (2015).CrossRefGoogle Scholar
[31] Wang, T.J. and Sun, T., Mixed pseudospectral method for heat transfer, Math. Mod. Anal. 21, 199–219 (2016).CrossRefGoogle Scholar
[32] Zhang, C. and Guo, B.Y., Domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations on unbounded domains, J. Sci. Comput. 53, 451–480 (2012).CrossRefGoogle Scholar
[33] Zhang, C. and Guo, B.Y., Generalized Hermite spectral method matching asymptotic behaviors. J. Comput. Appl. Math. 255, 616–634 (2014).CrossRefGoogle Scholar
[34] Xiang, X.M. and Wang, Z.Q., Generalized Hermite spectral method and its application to problems in unbounded domains, SIAM J. Numer. Anal. 48, 1231–1253 (2010).CrossRefGoogle Scholar