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January  2012, 17(1): 325-346. doi: 10.3934/dcdsb.2012.17.325

## Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains

 1 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

Received  August 2010 Revised  February 2011 Published  October 2011

In this paper, we develop several generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions for one/two dimensional Neumann problems. Some basic results on the generalized Jacobi rational approximations for Neumann problems are established, which play important roles in the related spectral methods. Three model problems are considered. The convergence of proposed schemes is proved. Numerical results demonstrate their spectral accuracy and efficiency.
Citation: Zhong-Qing Wang, Jing-Xia Wu. Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 325-346. doi: 10.3934/dcdsb.2012.17.325
##### References:
 [1] F. Auteri, N. Parolini and L. Quartapelle, Essential imposition of Neumann condition in Galerkin-Legendre elliptic solvers, J. Comput. Phys., 185 (2003), 427-444. doi: 10.1016/S0021-9991(02)00064-5. [2] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Spring-Verlag, Berlin-New York, 1976. [3] C. Bernardi and Y. Maday, Spectral Methods, in "Handbook of Numerical Analysis, Vol. V," 209-485, North-Holland, Amsterdam, 1997. [4] J. P. Boyd, Spectral method using rational basis functions on an infinite interval, J. Comp. Phys., 69 (1987), 112-142. doi: 10.1016/0021-9991(87)90158-6. [5] J. P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comp. Phys., 70 (1987), 63-88. doi: 10.1016/0021-9991(87)90002-7. [6] C. I. Christov, A complete orthonormal system of functions in $L^2(-\infty,\infty)$ space, SIAM J. Appl. Math., 42 (1982), 1337-1344. doi: 10.1137/0142093. [7] Ben-yu Guo, "Spectral Methods and Their Applications," World Scientific Publishing Co., Inc., River Edge, NJ, 1998. [8] Ben-yu Guo, Jie Shen and Li-lian Wang, Optional spectral-Galerkin methods using generalizd Jacobi polynomials, J. Sci. Comp., 27 (2006), 305-322. doi: 10.1007/s10915-005-9055-7. [9] Ben-yu Guo, Jie Shen and Li-lian Wang, Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math., 59 (2009), 1011-1028. doi: 10.1016/j.apnum.2008.04.003. [10] Ben-yu Guo, Jie Shen and Zhong-qing Wang, A rational approximation and its applications to differential equations on the half line, J. Sci. Comput., 15 (2000), 117-147. doi: 10.1023/A:1007698525506. [11] Ben-yu Guo, Jie Shen and Zhong-qing Wang, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Inter. J. Numer. Meth. Engin., 53 (2002), 65-84. doi: 10.1002/nme.392. [12] Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Appr. Theo., 128 (2004), 1-41. doi: 10.1016/j.jat.2004.03.008. [13] Ben-yu Guo and Tian-jun Wang, Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an finite channel, Math. Comp., 78 (2009), 129-151. doi: 10.1090/S0025-5718-08-02152-2. [14] Ben-yu Guo and Tian-jun Wang, Composite Laguerre-Legendre spectral method for exterior problems, Adv. Comput. Math., 32 (2010), 393-429. doi: 10.1007/s10444-008-9112-5. [15] Ben-yu Guo and Yong-gang Yi, Generalized Jacobi rational spectral method and its applications, J. Sci. Comput., 43 (2010), 201-238. doi: 10.1007/s10915-010-9353-6. [16] A. B. J. Kuijlaars, A. Martinez-Finkelshtein and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Elec. Tran. on Numer. Anal., 19 (2005), 1-17. [17] Li Huiyuan, "Super Spectral Viscosity Methods for Nonliear Conservation Laws, Chebyshev Collocation Methods and Their Applications," Ph.D Thesis, Shanghai University, Shanghai, China, 2002. [18] Jie 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. [19] Jie 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. [20] Jie Shen and Li-lian Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195-241. [21] G. Szegö, "Orthogonal Polynomials," Amer. Math. Soc., Providence, R.I., 1959. [22] Tian-jun Wang and Zhong-qing Wang, Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition, Appl. Numer. Math., 59 (2009), 2444-2451. doi: 10.1016/j.apnum.2009.05.003. [23] Zhong-qing Wang and Ben-yu Guo, Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp., 77 (2008), 883-907. [24] Zhong-qing Wang and Xu-hong Yu, Jacobi spectral method with essential imposition of Neumann boundary condition, submitted. [25] Yong-gang Yi and Ben-yu Guo, Generalized Jacobi rational spectral method and its applications to degenerated differentual equations on the half line, submitted.

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##### References:
 [1] F. Auteri, N. Parolini and L. Quartapelle, Essential imposition of Neumann condition in Galerkin-Legendre elliptic solvers, J. Comput. Phys., 185 (2003), 427-444. doi: 10.1016/S0021-9991(02)00064-5. [2] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Spring-Verlag, Berlin-New York, 1976. [3] C. Bernardi and Y. Maday, Spectral Methods, in "Handbook of Numerical Analysis, Vol. V," 209-485, North-Holland, Amsterdam, 1997. [4] J. P. Boyd, Spectral method using rational basis functions on an infinite interval, J. Comp. Phys., 69 (1987), 112-142. doi: 10.1016/0021-9991(87)90158-6. [5] J. P. Boyd, Orthogonal rational functions on a semi-infinite interval, J. Comp. Phys., 70 (1987), 63-88. doi: 10.1016/0021-9991(87)90002-7. [6] C. I. Christov, A complete orthonormal system of functions in $L^2(-\infty,\infty)$ space, SIAM J. Appl. Math., 42 (1982), 1337-1344. doi: 10.1137/0142093. [7] Ben-yu Guo, "Spectral Methods and Their Applications," World Scientific Publishing Co., Inc., River Edge, NJ, 1998. [8] Ben-yu Guo, Jie Shen and Li-lian Wang, Optional spectral-Galerkin methods using generalizd Jacobi polynomials, J. Sci. Comp., 27 (2006), 305-322. doi: 10.1007/s10915-005-9055-7. [9] Ben-yu Guo, Jie Shen and Li-lian Wang, Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math., 59 (2009), 1011-1028. doi: 10.1016/j.apnum.2008.04.003. [10] Ben-yu Guo, Jie Shen and Zhong-qing Wang, A rational approximation and its applications to differential equations on the half line, J. Sci. Comput., 15 (2000), 117-147. doi: 10.1023/A:1007698525506. [11] Ben-yu Guo, Jie Shen and Zhong-qing Wang, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Inter. J. Numer. Meth. Engin., 53 (2002), 65-84. doi: 10.1002/nme.392. [12] Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Appr. Theo., 128 (2004), 1-41. doi: 10.1016/j.jat.2004.03.008. [13] Ben-yu Guo and Tian-jun Wang, Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an finite channel, Math. Comp., 78 (2009), 129-151. doi: 10.1090/S0025-5718-08-02152-2. [14] Ben-yu Guo and Tian-jun Wang, Composite Laguerre-Legendre spectral method for exterior problems, Adv. Comput. Math., 32 (2010), 393-429. doi: 10.1007/s10444-008-9112-5. [15] Ben-yu Guo and Yong-gang Yi, Generalized Jacobi rational spectral method and its applications, J. Sci. Comput., 43 (2010), 201-238. doi: 10.1007/s10915-010-9353-6. [16] A. B. J. Kuijlaars, A. Martinez-Finkelshtein and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Elec. Tran. on Numer. Anal., 19 (2005), 1-17. [17] Li Huiyuan, "Super Spectral Viscosity Methods for Nonliear Conservation Laws, Chebyshev Collocation Methods and Their Applications," Ph.D Thesis, Shanghai University, Shanghai, China, 2002. [18] Jie 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. [19] Jie 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. [20] Jie Shen and Li-lian Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195-241. [21] G. Szegö, "Orthogonal Polynomials," Amer. Math. Soc., Providence, R.I., 1959. [22] Tian-jun Wang and Zhong-qing Wang, Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition, Appl. Numer. Math., 59 (2009), 2444-2451. doi: 10.1016/j.apnum.2009.05.003. [23] Zhong-qing Wang and Ben-yu Guo, Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp., 77 (2008), 883-907. [24] Zhong-qing Wang and Xu-hong Yu, Jacobi spectral method with essential imposition of Neumann boundary condition, submitted. [25] Yong-gang Yi and Ben-yu Guo, Generalized Jacobi rational spectral method and its applications to degenerated differentual equations on the half line, submitted.
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