Spectral collocation method for convection-diffusion equation

• Autorzy: Li Jin , Cheng Yongling

Identyfikatory

DOI

10.1515/dema-2023-0110

• Języki publikacji: EN

Abstrakty

EN

Spectral collocation method, named linear barycentric rational interpolation collocation method (LBRICM), for convection-diffusion (C-D) equation with constant coefficient is considered. We change the discrete linear equations into the matrix equation. Different from the classical methods to solve the C-D equation, we solve the C-D equation with the time variable and space variable obtained at the same time. Furthermore, the convergence rate of the C-D equation by LBRICM is proved. Numerical examples are presented to test our analysis.

Słowa kluczowe

EN

linear rational interpolation; convection-diffusion equation; Barycentric form; error estimate; matrix equation

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20230110
• Opis fizyczny: Bibliogr. 38 poz., rys., tab.

Twórcy

autor

Li Jin

• School of Science, Shandong Jianzhu University, Jinan, 250100, P. R. China

autor

Cheng Yongling

• School of Science, Shandong Jianzhu University, Jinan, 250100, P. R. China

Bibliografia

• [1] N. A. Kudryashov and D. I. Sineshchikov, Analytical solution of a nonlinear convection-diffusion equation with polynomial sources, Model. Anal. Inform. Sist. 23 (2016), 309–316.
• [2] J. Shen, T. Tang, and L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer, London, 2011.
• [3] H. Wang, B. Shi, H. Liang, and Z. Chai, Finite difference lattice Boltzmann modal for nonlinear convection-diffusion equations, Appl. Math. Comput. 309 (2017), 334–349, DOI: https://doi.org/10.1016/j.amc.2017.04.015.
• [4] R. Eymard, T. Gallonet, and R. Herbin, Error estimate for approximate solutions of a nonlinear convection-diffusion problem, Adv. Differential Equations 7 (2002), 419–440, DOI: https://doi.org/10.1093/imanum/18.4.563.
• [5] J. D. Frutos, B. G. Archilla, and J. Novo, Nonlinear convection-diffusion problems: fully discrete approximations and a posteriori error estimates, IMA J. Numer. Anal. 31 (2011), 1402–1430, DOI: https://doi.org/10.1093/imanum/drq017.
• [6] C. Canuto and A. Quarteroni, Error estimates for spectral and pseudospectral approximations of hyperbolic equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 629–642.
• [7] A. Mohebbi and M. Dehghan. High-order compact solution of the one-dimensional heat and advection-diffusion equations, Appl. Math. Model. 34 (2010), 3071–3084, DOI: https://doi.org/10.1016/j.apm.2010.01.013.
• [8] W. M. Abd-Elhameed and Y. H. Youssri, New formulas of the high-order derivatives of fifth-kind Chebyshev polynomials: Spectral solution of the convection-diffusion equation, Numer. Methods Partial Differential Equations (2023), 1–17, DOI: https://doi.org/10.1002/num.22756.
• [9] N. R. Bayramov and J. K. Kraus, On the stable solution of transient convection-diffusion equations, J. Comput. Appl. Math. 280 (2015), 275–293, DOI: https://doi.org/10.1016/j.cam.2014.12.001.
• [10] Y. H. Youssri and A. G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional diffusion equation, Contemp. Math. [Internet], 4(2023 Apr. 8), no. 2, 230–248, [cited 2023 Jun. 22], https://ojs.wiserpub.com/index.php/CM/article/view/2420.
• [11] M. Moustafa, Y. H. Youssri, and A. G. Atta, Explicit Chebyshev-Galerkin scheme for the time-fractional diffusion equation, J. Modern Phys. C. 35 (2024), 2450002, DOI: https://doi.org/10.1142/S0129183124500025.
• [12] A. G. Atta, W. M. Abd-Elhameed, and Y. H. Youssr, Shifted fifth-kind Chebyshev polynomials Galerkin-based procedure for treating fractional diffusion-wave equation, Int. J. Modern Phys. C 33 (2022), no. 8, 2250102, DOI: https://doi.org/10.1142/S0129183122501029.
• [13] M. A. Abdelkawy and S. A. Alyami, Legendre-Chebyshev spectral collocation method for two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional, Chaos Solitons Fractals 151 (2021), 111279, DOI: https://doi.org/10.1016/j.chaos.2021.111279.
• [14] V. F. Morales-Delgado, J. F. Gómez-Aguilar, K. M. Saad, M. A. Khan, and P. Agarwal, Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach, Physica A Stat. Mech. Appl. 523 (2019), 48–65, DOI: https://doi.org/10.1016/j.physa.2019.02.018.
• [15] P. Berrut, M. S. Floater, and G. Klein, Convergence rates of derivatives of a family of barycentric rational interpolants, Appl. Numer. Math. 61 (2011), no. 9, 989–1000, DOI: https://doi.org/10.1016/j.apnum.2011.05.001.
• [16] P. Berrut and G. Klein, Recent advances in linear barycentric rational interpolation, J. Comput. Appl. Math. 259 (2014), Part A, 95–107, DOI: https://doi.org/10.1016/j.cam.2013.03.044.
• [17] G. Klein and J. P. Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, Siam J. Numer. Anal. 50 (2012), no. 2, 643–656, DOI: https://doi.org/10.1137/110827156.
• [18] G. Klein and J.-P. Berrut, Linear barycentric rational quadrature, BIT Numer. Math. 52 (2012), 407–424, DOI: https://doi.org/10.1007/s10543-011-0357-x.
• [19] M. S. Floater and K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107 (2007), no. 2, 315–331, DOI: https://doi.org/10.1007/s00211-007-0093-y.
• [20] A. Abdi, J-P. Berrut, and S. A. Hosseini, The linear barycentric rational method for a class of delay Volterra integro-differential equations, J. Scientific Comput. 75 (2001), no. 3, 1195–1210, DOI: https://doi.org/10.1007/s10915-017-0608-3.
• [21] S. Li and Z. Wang, High Precision Meshless Barycentric Interpolation Collocation Method-Algorithmic Program and Engineering Application, Science Publishing, 2012.
• [22] J. P. Berrut, S. A. Hosseini, and G. Klein, The linear barycentric rational quadrature method for Volterra integral equations, SIAM J. Sci. Comput. 36 (2014), no. 1, 105–123, DOI: https://doi.org/10.1137/120904020.
• [23] M. Li and C. Huang, The linear barycentric rational quadrature method for auto-convolution Volterra integral equations, J. Sci. Comput. 78 (2019), no. 1, 549–564, DOI: https://doi.org/10.1007/s10915-018-0779-6.
• [24] J. Leey and L. Greengardz, A fast adaptive numerical method for stiff two-point boundary value problems, Siam. J. Sci. Comput. 18 (1997), no. 2, 403–429, DOI: https://doi.org/10.1137/S1064827594272797.
• [25] Z. Wang, L. Zhang, Z. Xu, and J. Li, Barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems, Chinese J. Appl. Mechanics 35 (2018), no. 2, 304–309.
• [26] Z. Wang, Z. Xu, and J. Li, Mixed barycentric interpolation collocation method of displacement-pressure for incompressible plane elastic problems, Chinese J. Appl. Mechanics. 35 (2018), no. 3, 195–201.
• [27] Z. Wang and S. Li, Barycentric Interpolation Collocation Method for Nonlinear Problems, National Defense Industry Press, Beijing, 2015.
• [28] E. Cirillo and K. Hormann, On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes, J. Comput. Appl. Math. 349 (2019), 292–301, DOI: https://doi.org/10.1016/j.cam.2018.06.011.
• [29] M. S. Floater and H. Kai Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math. 107 (2007), no. 2, 315–331, DOI: https://doi.org/10.1007/s00211-007-0093-y.
• [30] J. Li, Linear barycentric rational collocation method for solving biharmonic equation, Demonstr. Math. 55 (2022), 587–603, DOI: https://doi.org/10.1515/dema-2022-0151.
• [31] J. Li, Y. Cheng, Z. C. Li, and Z. K. Tian, Linear barycentric rational collocation method for solving generalized Poisson equations, Math. Biosci. Eng. 20 (2023), no. 3, 4782–4797, DOI: https://doi.org/10.3934/mbe.2023221.
• [32] J. Li, X. Su, and K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simulat. 205 (2023), 340–367, DOI: https://doi.org/10.1016/j.matcom.2022.10.005.
• [33] J. Li and Y. Cheng, Barycentric rational interpolation method for solving fractional cable equation, Electr. Res. Archive 31 (2023), no. 6, 3649–3665, DOI: https://doi.org/10.3934/era.2023185.
• [34] J. Li, Barycentric rational collocation method for fractional reaction-diffusion equation, AIMS Math. 8 (2023), no. 4, 9009–9026, DOI: https://doi.org/10.3934/math.2023451.
• [35] J. Li, X. Su, and J. Qu, Linear barycentric rational collocation method for solving telegraph equation, Math. Meth. Appl. Sci. 44 (2021), 11720–11737, DOI: https://doi.org/10.1002/mma.7548.
• [36] J. Li and Y. Cheng. Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation, Comput. Appl. Math. 39 (2020), 1–9, DOI: https://doi.org/10.1007/s40314-020-1114-z.
• [37] J. Li and Y. Cheng, Linear barycentric rational collocation method for solving heat conduction equation, Numer. Methods Partial Differential Equations 37 (2021), no. 1, 533–545, DOI: https://doi.org/10.1002/num.22539.
• [38] G. W. Stewart, Matrix Algorithms, Vol. II, Eigensystems Paperback-August 1, Philadephia, 2001.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).