A Legendre-tau-Galerkin method in time for two-dimensional Sobolev equations

Siqin Tang; Hong Li; Siqin Tang; Hong Li

doi:10.3934/math.2023820

AIMS Mathematics

2023, Volume 8, Issue 7: 16073-16093. doi: 10.3934/math.2023820

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Research article

A Legendre-tau-Galerkin method in time for two-dimensional Sobolev equations

Siqin Tang ^1,2,
Hong Li ^{1
,
,}

1.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
2.
Faculty of Science, Inner Mongolia University of Technology, Hohhot 010015, China

Received: 10 January 2023 Revised: 10 April 2023 Accepted: 12 April 2023 Published: 05 May 2023
MSC : 65M12, 65M70

This work is devoted to present the Legendre space-time spectral method for two-dimensional (2D) Sobolev equations. Considering the asymmetry of the first-order differential operator, the Legendre-tau-Galerkin method is employed in time discretization and its multi-interval form is also investigated. In the theoretical analysis, rigorous proof of the stability and $ L^2(\Sigma) $-error estimates is given for the fully discrete schemes in both single-interval and multi-interval forms. Being different from the general Legendre-Galerkin method, we specifically take the Fourier-like basis functions in space to save the computing time and memory in the algorithm of the proposed method. Numerical experiments were included to confirm that our method attains exponential convergence in both time and space and that the multi-interval form can achieve improved numerical results compared with the single interval form.
- 2D Sobolev equations,
- Fourier-like basis functions,
- Legendre space-time spectral method,
- exponential convergence
Citation: Siqin Tang, Hong Li. A Legendre-tau-Galerkin method in time for two-dimensional Sobolev equations[J]. AIMS Mathematics, 2023, 8(7): 16073-16093. doi: 10.3934/math.2023820

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Abstract

This work is devoted to present the Legendre space-time spectral method for two-dimensional (2D) Sobolev equations. Considering the asymmetry of the first-order differential operator, the Legendre-tau-Galerkin method is employed in time discretization and its multi-interval form is also investigated. In the theoretical analysis, rigorous proof of the stability and $ L^2(\Sigma) $-error estimates is given for the fully discrete schemes in both single-interval and multi-interval forms. Being different from the general Legendre-Galerkin method, we specifically take the Fourier-like basis functions in space to save the computing time and memory in the algorithm of the proposed method. Numerical experiments were included to confirm that our method attains exponential convergence in both time and space and that the multi-interval form can achieve improved numerical results compared with the single interval form.

References

[1]	G. Mesri, A. Rokhsar, Theory of consolidation for clays, J. Geotech. Eng. Div., 100 (1974), 889–904. https://doi.org/10.1061/AJGEB6.0000075 doi: 10.1061/AJGEB6.0000075
[2]	P. J. Chen, M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys., 19 (1968), 614–627. https://doi.org/10.1007/BF01594969 doi: 10.1007/BF01594969
[3]	X. Cao, I. S. Pop, Degenerate two-phase porous media flow model with dynamic capillarity, J. Differ. Equ., 260 (2016), 2418–2456. https://doi.org/10.1016/j.jde.2015.10.008 doi: 10.1016/j.jde.2015.10.008
[4]	Ankur, R. Jiwari, N. Kumar, Analysis and simulation of Korteweg-de Vries-Rosenau-regularised long-wave model via Galerkin finite element method, Comput. Math. Appl., 135 (2023), 134–148. https://doi.org/10.1016/j.camwa.2023.01.027 doi: 10.1016/j.camwa.2023.01.027
[5]	Z. C. Fang, J. Zhao, H. Li, Y. Liu, A fast time two-mesh finite volume element algorithm for the nonlinear time-fractional coupled diffusion model, Numer. Algorithms, 2022 (2022), 1–36. https://doi.org/10.1007/s11075-022-01444-2 doi: 10.1007/s11075-022-01444-2
[6]	K. H. Kumar, R. Jiwari, A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution, Int. J. Comput. Math., 99 (2022), 2221–2236. https://doi.org/10.1080/00207160.2022.2041193 doi: 10.1080/00207160.2022.2041193
[7]	Y. X. Niu, Y. Liu, H. Li, F. W. Liu, Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media, Math. Comput. Simul., 203 (2023), 387–407. https://doi.org/10.1016/j.matcom.2022.07.001 doi: 10.1016/j.matcom.2022.07.001
[8]	H. Li, Z. D. Luo, J. An, P. Sun, A fully discrete finite volume element formulation for Sobolev equation and numerical simulations, Math. Numer. Sinica, 34 (2012), 163–172. https://doi.org/10.12286/jssx.2012.2.163 doi: 10.12286/jssx.2012.2.163
[9]	Z. D. Luo, F. Teng, J. Chen, A POD-based reduced-order Crank-Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations, Math. Comput. Simul., 146 (2018), 118–133. https://doi.org/10.1016/j.matcom.2017.11.002 doi: 10.1016/j.matcom.2017.11.002
[10]	Z. D. Luo, A Crank-Nicolson finite volume element method for two-dimensional Sobolev equations, J. Inequal. Appl., 2016 (2016), 1–15. https://doi.org/10.1186/s13660-016-1131-z doi: 10.1186/s13660-016-1131-z
[11]	X. Q. Zhang, W. Q. Wang, T. C. Lu, Continuous interior penalty finite element methods for Sobolev equations with convection-dominated term, Numer. Methods Partial Differ. Equ., 28 (2012), 1399–1416. https://doi.org/10.1002/num.20693 doi: 10.1002/num.20693
[12]	Z. H. Zhao, H. Li, Z. D. Luo, Analysis of a space-time continuous Galerkin method for convection-dominated Sobolev equations, Comput. Math. Appl., 73 (2017), 1643–1656. https://doi.org/10.1016/j.camwa.2017.01.023 doi: 10.1016/j.camwa.2017.01.023
[13]	T. J. Sun, D. P. Yang, The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term, Appl. Math. Comput., 125 (2002), 325–345. https://doi.org/10.1016/S0096-3003(00)00135-1 doi: 10.1016/S0096-3003(00)00135-1
[14]	M. Abbaszadeh, M. Dehghan, Interior penalty discontinuous Galerkin technique for solving generalized Sobolev equation, Appl. Numer. Math., 154 (2020), 172–186. https://doi.org/10.1016/j.apnum.2020.03.019 doi: 10.1016/j.apnum.2020.03.019
[15]	D. Y. Shi, J. J. Sun, Superconvergence analysis of an $H^1$-Galerkin mixed finite element method for Sobolev equations, Comput. Math. Appl., 72 (2016), 1590–1602. https://doi.org/10.1016/j.camwa.2016.07.023 doi: 10.1016/j.camwa.2016.07.023
[16]	X. L. Li, H. X. Rui, A block-centered finite difference method for the nonlinear Sobolev equation on nonuniform rectangular grids, Appl. Math. Comput., 363 (2019), 124607. https://doi.org/10.1016/j.amc.2019.124607 doi: 10.1016/j.amc.2019.124607
[17]	S. He, H. Li, Y. Liu, Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations, Front. Math. China, 8 (2013), 825–836. https://doi.org/10.1007/s11464-013-0307-9 doi: 10.1007/s11464-013-0307-9
[18]	M. Dehghan, N. Shafieeabyaneh, M. Abbaszadeh, Application of spectral element method for solving Sobolev equations with error estimation, Appl. Numer. Math., 158 (2020), 439–462. https://doi.org/10.1016/j.apnum.2020.08.010 doi: 10.1016/j.apnum.2020.08.010
[19]	A. Quarteroni, Fourier spectral methods for pseudoparabolic equations, SIAM J. Numer. Anal., 24 (1987), 323–335. https://doi.org/10.1137/0724024 doi: 10.1137/0724024
[20]	C. Zhang, H. F. Yao, H. Y. Li, New space-time spectral and structured spectral element methods for high order problems, J. Comput. Appl. Math., 351 (2019), 153–166. https://doi.org/10.1016/j.cam.2018.08.038 doi: 10.1016/j.cam.2018.08.038
[21]	J. Scheffel, K. Lindvall, H. F. Yik, A time-spectral approach to numerical weather prediction, Comput. Phys. Commun., 226 (2018), 127–135. https://doi.org/10.1016/j.cpc.2018.01.010 doi: 10.1016/j.cpc.2018.01.010
[22]	Y. H. Qin, H. P. Ma, Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations, Appl. Numer. Math., 153 (2020), 52–65. https://doi.org/10.1016/j.apnum.2020.02.001 doi: 10.1016/j.apnum.2020.02.001
[23]	S. H. Lui, Legendre spectral collocation in space and time for PDEs, Numer. Math., 136 (2017), 75–99. https://doi.org/10.1007/s00211-016-0834-x doi: 10.1007/s00211-016-0834-x
[24]	S. H. Lui, S. Nataj, Spectral collocation in space and time for linear PDEs, J. Comput. Phys., 424 (2021), 109843. https://doi.org/10.1016/j.jcp.2020.109843 doi: 10.1016/j.jcp.2020.109843
[25]	W. J. Liu, J. B. Sun, B. Y. Wu, Space-time spectral method for the two-dimensional generalized sine-Gordon equation, J. Math. Anal. Appl., 427 (2015), 787–804. https://doi.org/10.1016/j.jmaa.2015.02.057 doi: 10.1016/j.jmaa.2015.02.057
[26]	S. Q. Tang, H. Li, B. L. Yin, A space-time spectral method for multi-dimensional Sobolev equations, J. Math. Anal. Appl., 499 (2021), 124937. https://doi.org/10.1016/j.jmaa.2021.124937 doi: 10.1016/j.jmaa.2021.124937
[27]	J. G. Tang, H. P. Ma, Single and multi-interval Legendre $\tau$-methods in time for parabolic equations, Adv. Comput. Math., 17 (2002), 349–367. https://doi.org/10.1023/A:1016273820035 doi: 10.1023/A:1016273820035
[28]	J. G. Tang, H. P. Ma, Single and multi-interval Legendre spectral methods in time for parabolic equations, Numer. Methods Partial Differ. Equ., 22 (2006), 1007–1034. https://doi.org/10.1002/num.20135 doi: 10.1002/num.20135
[29]	J. G. Tang, H. P. Ma, A Legendre spectral method in time for first-order hyperbolic equations, Appl. Numer. Math., 57 (2007), 1–11. https://doi.org/10.1016/j.apnum.2005.11.009 doi: 10.1016/j.apnum.2005.11.009
[30]	J. Shen, L. L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710–720. https://doi.org/10.1016/j.apnum.2006.07.012 doi: 10.1016/j.apnum.2006.07.012
[31]	J. Shen, T. Tang, L. L. Wang, Spectral methods: Algorithms, analysis and applications, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7
[32]	A. J. Laub, Matrix analysis for scientists and engineers, Philadelphia: SIAM, 2004.

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