Analysis of a fourth-order compact $ \theta $-method for delay parabolic equations

Lili Li; Boya Zhou; Huiqin Wei; Fengyan Wu; Lili Li; Boya Zhou; Huiqin Wei; Fengyan Wu

doi:10.3934/era.2024127

Electronic Research Archive

2024, Volume 32, Issue 4: 2805-2823. doi: 10.3934/era.2024127

Previous Article Next Article

Research article Special Issues

Analysis of a fourth-order compact $ \theta $-method for delay parabolic equations

1.
Department of Public Basic Teaching and Research, Shandong Police College, Jinan 250014, China
2.
School of Mathematics and Big Data, Foshan University, Guangdong 52800, China
3.
School of Information Science and Technology, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
4.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received: 23 January 2024 Revised: 09 March 2024 Accepted: 18 March 2024 Published: 09 April 2024

The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the nth power of the iteration matrix derived via a fourth-order compact $ \theta $-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.
- delay parabolic equations,
- high-order,
- numerical stability,
- error bounds,
- compact $ \theta $-method
Citation: Lili Li, Boya Zhou, Huiqin Wei, Fengyan Wu. Analysis of a fourth-order compact $ \theta $-method for delay parabolic equations[J]. Electronic Research Archive, 2024, 32(4): 2805-2823. doi: 10.3934/era.2024127

Related Papers:

Abstract

The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the nth power of the iteration matrix derived via a fourth-order compact $ \theta $-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.

References

[1]	H. Gong, C. Wang, X. Zhang, Partial regularity of suitable weak solutions of the Navier-Stokes-Planck-Nernst-Poisson equation, SIAM J. Math. Anal., 53 (2021), 3306–3337. https://doi.org/10.1137/19M1292011 doi: 10.1137/19M1292011
[2]	X. Zhai, Y. Chen, Y. Li, Large global solutions of the compressible Navier-Stokes equations in three dimensions, Discrete Contin. Dyn. Syst.: Ser. A, 43 (2023), 309–337. https://doi.org/10.3934/dcds.2022150 doi: 10.3934/dcds.2022150
[3]	Y. Chen, F. Zou, Nonlinear stability of strong traveling waves for a chemotaxis model with logarithmic sensitivity and periodic perturbations, Math. Methods Appl. Sci., 46 (2023), 15123–15146. https://doi.org/10.1002/mma.9365 doi: 10.1002/mma.9365
[4]	F. Liu, J. Yang, X. Yu, Positive solutions to multi-critical elliptic problems, Ann. Mat. Pura Appl., 202 (2023), 851–875. https://doi.org/10.1007/s10231-022-01262-2 doi: 10.1007/s10231-022-01262-2
[5]	B. Dong, J. Wu, X. Zhai, Global small solutions to a special 212-D compressible viscous non-resistive MHD system, J. Nonlinear Sci., 33 (2023). https://doi.org/10.1007/s00332-022-09881-y doi: 10.1007/s00332-022-09881-y
[6]	F. R. Lin, X. Q. Jin, S. L. Lei, Strang-type preconditioners for solving linear systems from delay differential equations, BIT Numer. Math., 43 (2003), 139–152. https://doi.org/10.1007/s100920300001 doi: 10.1007/s100920300001
[7]	Q. Q. Tian, H. X. Zhang, X. H. Yang, X. X. Jiang, An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel, Comput. Appl. Math., 41 (2022), 328. https://doi.org/10.1007/s40314-022-02040-9 doi: 10.1007/s40314-022-02040-9
[8]	Y. Shi, X. Yang, Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation, Electron. Res. Arch., 32 (2024), 1471–1497. https://doi.org/10.3934/era.2024068 doi: 10.3934/era.2024068
[9]	J. W. Wang, X. X. Jiang, H. X. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
[10]	Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., (2024), 1–28.
[11]	H. X. Zhang, Y. Liu, X. H. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674. https://doi.org/10.1007/s12190-022-01760-9 doi: 10.1007/s12190-022-01760-9
[12]	Z. Y. Zhou, H. X. Zhang, X. H. Yang, J. Tang, An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions, Int. J. Comput. Math., 100 (2023), 1719–1736. https://doi.org/10.1080/00207160.2023.2212307 doi: 10.1080/00207160.2023.2212307
[13]	H. X. Zhang, X. X. Jiang, F. R. Wang, X. H. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., (2024), 1–25. https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
[14]	G. Yuan, D. Ding, W. Lu, F. Wu, A linearized fourth-order compact ADI method for phytoplankton-zooplankton model arising in marine ecosystem, Comput. Appl. Math., 43 (2024), 1–22. https://doi.org/10.1007/s40314-023-02570-w doi: 10.1007/s40314-023-02570-w
[15]	Y. Kuang, Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
[16]	J. Kongson, S. Amornsamankul, A model of the signal transduction process under a delay, East Asian J. Appl. Math., 7 (2017), 741–751. https://doi.org/10.4208/eajam.181016.300517a doi: 10.4208/eajam.181016.300517a
[17]	W. Kang, E. Fridman, Boundary constrained control of delayed nonlinear Schrödinger equation, IEEE Trans. Autom. Control, 63 (2018), 3873–3880. https://doi.org/10.1109/TAC.2018.2800526 doi: 10.1109/TAC.2018.2800526
[18]	T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072. https://doi.org/10.1016/j.aml.2019.106072 doi: 10.1016/j.aml.2019.106072
[19]	T. Zhang, Y. Li, Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
[20]	J. Wu, Theory and Application of Partial Functional Differential Equation, in Applied Mathematical Sciences, Springer, 1996.
[21]	A. Bellen, M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
[22]	D. Li, C. Zhang, W. Wang, Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Anal. Real. World Appl., 13 (2012), 1401–1415. https://doi.org/10.1016/j.nonrwa.2011.11.005 doi: 10.1016/j.nonrwa.2011.11.005
[23]	C. Tang, C. Zhang, A fully discrete $\theta$-method for solving semi-linear reaction-diffusion equations with time-variable delay, Math. Comput. Simulat., 179 (2021), 48–56. https://doi.org/10.1016/j.matcom.2020.07.019 doi: 10.1016/j.matcom.2020.07.019
[24]	J. Xie, Z. Zhang, The high-order multistep ADI solver for two-dimensional nonlinear delayed reaction-diffusion equations with variable coefficients, Comput. Math. Appl., 75 (2018), 3558–3570. https://doi.org/10.1016/j.camwa.2018.02.017 doi: 10.1016/j.camwa.2018.02.017
[25]	J. Xie, D. Deng, H. Zheng, A compact difference scheme for one-dimensional nonlinear delay reaction-diffusion equations with variable coefficient, IAENG Int. J. Appl. Math., 47 (2017), 14–19.
[26]	T. Zhang, Y. Liu, Global mean-square exponential stability and random periodicity of discrete-time stochastic inertial neural networks with discrete spatial diffusions and Dirichlet boundary condition, Comput. Math. Appl., 141 (2023), 116–128. https://doi.org/10.1016/j.camwa.2023.04.011 doi: 10.1016/j.camwa.2023.04.011
[27]	H. Liang, Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delay, Appl. Math. Comput., 15 (2015), 160–178. https://doi.org/10.1016/j.amc.2015.04.104 doi: 10.1016/j.amc.2015.04.104
[28]	G. Zhang, A. Xiao, J. Zhou, Implicit-explicit multistep finite-element methods for nonlinear convection-diffusion-reaction equations with time delay, Int. J. Comput. Math., 95 (2018), 2496–2510. https://doi.org/10.1080/00207160.2017.1408802 doi: 10.1080/00207160.2017.1408802
[29]	W. Wang, L. Yi, A. Xiao, A posteriori error estimates for fully discrete finite element method for generalized diffusion equation with delay, J. Sci. Comput., 84 (2020), 1–27. https://doi.org/10.1007/s10915-020-01262-5 doi: 10.1007/s10915-020-01262-5
[30]	H. Han, C. Zhang, Galerkin finite element methods solving 2D initial-boundary value problems of neutral delay-reaction-diffusion equations, Comput. Math. Appl., 92 (2021), 159–171. https://doi.org/10.1016/j.camwa.2021.03.030 doi: 10.1016/j.camwa.2021.03.030
[31]	X. H. Yang, W. L. Qiu, H. F. Chen, H. X. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
[32]	D. Li, C. Zhang, Nonlinear stability of discontinuous Galerkin methods for delay differential equations, Appl. Math. Lett., 23 (2010), 457–461. https://doi.org/10.1016/j.aml.2009.12.003 doi: 10.1016/j.aml.2009.12.003
[33]	D. Li, C. Zhang, Superconvergence of a discontinuous Galerkin method for first-order linear delay differential equations, J. Comput. Math., (2011), 574–588. https://doi.org/10.4208/jcm.1107-m3433 doi: 10.4208/jcm.1107-m3433
[34]	D. Li, C. Zhang, $L^\infty$ error estimates of discontinuous Galerkin methods for delay differential equations, Appl. Numer. Math., 82 (2014), 1–10. https://doi.org/10.1016/j.apnum.2014.01.008 doi: 10.1016/j.apnum.2014.01.008
[35]	G. Zhang, X. Dai, Superconvergence of discontinuous Galerkin method for neutral delay differential equations, Int. J. Comput. Math., 98 (2021), 1648–1662. https://doi.org/10.1080/00207160.2020.1846030 doi: 10.1080/00207160.2020.1846030
[36]	A. Araújo, J. R. Branco, J. A. Ferreira, On the stability of a class of splitting methods for integro-differential equations, Appl. Numer. Math., 59 (2009), 436–453. https://doi.org/10.1016/j.apnum.2008.03.005 doi: 10.1016/j.apnum.2008.03.005
[37]	X. H. Yang, Z. M. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
[38]	X. H. Yang, H. X. Zhang, Q. Zhang, G. Y. Yuan, Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear Dyn., 108 (2022), 3859–3886. https://doi.org/10.1007/s11071-022-07399-2 doi: 10.1007/s11071-022-07399-2
[39]	E. Ávila-Vales, Á. G. C. Pérez, Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment, Chaos Soliton Fract., 127 (2019), 55–69. https://doi.org/10.1016/j.chaos.2019.06.024 doi: 10.1016/j.chaos.2019.06.024
[40]	H. Akca, G. E. Chatzarakis, I. P. Stavroulakis, An oscillation criterion for delay differential equations with several non-monotone arguments, Appl. Math. Lett., 59 (2016), 101–108. https://doi.org/10.1016/j.aml.2016.03.013 doi: 10.1016/j.aml.2016.03.013
[41]	J. Zhao, Y. Li, Y. Xu, Convergence and stability analysis of exponential general linear methods for delay differential equations, Numer. Math. Theory Methods Appl., 11 (2018), 354–382. https://doi.org/10.4208/nmtma.OA-2017-0032 doi: 10.4208/nmtma.OA-2017-0032
[42]	A. S. Hendy, V. G. Pimenov, J. E. Macias-Diaz, Convergence and stability estimates in difference setting for time-fractional parabolic equations with functional delay, Numer. Methods Part. Differ. Equations, 36 (2020), 118–132. https://doi.org/10.1002/num.22421 doi: 10.1002/num.22421
[43]	L. Blanco-Cocom, E. Ávila-Vales, Convergence and stability analysis of the $\theta$-method for delayed diffusion mathematical models, Appl. Math. Comput., 231 (2014), 16–26. https://doi.org/10.1016/j.amc.2013.12.188 doi: 10.1016/j.amc.2013.12.188
[44]	L. J. Wu, H. X. Zhang, X. H. Yang, The finite difference method for the fourth-order partial integro-differential equations with the multi-term weakly singular kernel, Math. Method Appl. Sci., 46 (2023), 2517–2537. https://doi.org/10.1002/mma.8658 doi: 10.1002/mma.8658
[45]	L. J. Wu, H. X. Zhang, X. H.Yang, F. R. Wang, A second-order finite difference method for the multi-term fourth-order integral-differential equations on graded meshes, Comput. Appl. Math., 41 (2022), 313. https://doi.org/10.1007/s40314-022-02026-7 doi: 10.1007/s40314-022-02026-7
[46]	C. Huang, S. Vandewalle, Unconditionally stable difference methods for delay partial differential equations, Numer. Math., 122 (2012), 579–601. https://doi.org/10.1007/s00211-012-0467-7 doi: 10.1007/s00211-012-0467-7
[47]	D. Li, C. Zhang, J. Wen, A note on compact finite difference method for reaction-diffusion equations with delay, Appl. Math. Model., 39 (2015), 1749–1754. https://doi.org/10.1016/j.apm.2014.09.028 doi: 10.1016/j.apm.2014.09.028
[48]	D. Green, H. W. Stech, Diffusion and Hereditary Effects in a Class of Population Models in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, Academic Press, New York, 1981.
[49]	Q. Zhang, M. Chen, Y. Xu, D. Xu, Compact $\theta$-method for the generalized delay diffusion equation, Appl. Math. Comput., 316 (2018), 357–369. https://doi.org/10.1016/j.amc.2017.08.033 doi: 10.1016/j.amc.2017.08.033
[50]	F. Wu, D. Li, J. Wen, J. Duan, Stability and convergence of compact finite difference method for parabolic problems with delay, Appl. Math. Comput., 322 (2018), 129–139. https://doi.org/10.1016/j.amc.2017.11.032 doi: 10.1016/j.amc.2017.11.032
[51]	H. Tian, Asymptotic stability analysis of the linear $\theta$-method for linear parabolic differential equations with delay, J. Differ. Equations. Appl., 15 (2009), 473–487. https://doi.org/10.1080/10236190802128284 doi: 10.1080/10236190802128284
[52]	S. V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an overdetermined problem, Numer. Math., 4 (1962), 277–292.
[53]	M. N. Spijker, Numerical stability, resolvent conditions and delay differential equations, Appl. Numer. Math., 24 (1997), 233–246. https://doi.org/10.1016/S0168-9274(97)00023-8 doi: 10.1016/S0168-9274(97)00023-8
[54]	J. van Dorsselaer, J. Kraaijevanger, M. N. Spijker, Linear stability analysis in the numerical solution of initial value problems, Acta Numer., (1993), 199–237.
[55]	B. Zubik-Kowal, Stability in the numerical solution of linear parabolic equations with a delay term, BIT Numer. Math., 41 (2001), 191–206. https://doi.org/10.1023/A:1021930104326 doi: 10.1023/A:1021930104326
[56]	E. G. Van den Heuvel, Using resolvent conditions to obtain new stability results for $\theta$-methods for delay differential equations, IMA J. Numer. Anal., 1 (2001), 421–438. https://doi.org/10.1093/imanum/21.1.421 doi: 10.1093/imanum/21.1.421
[57]	S. K. Jaffer, J. Zhao, M. Liu, Stability of linear multistep methods for delay differential equations in the light of Kreiss resolvent condition, Journal of Harbin Insititute of Technology-English edition, 8 (2001), 155–158.
[58]	C. Lubich, O. Nevanlinna, On resolvent conditions and stability estimates, BIT Numer. Math., 31 (1991), 293–313. https://doi.org/10.1007/BF01931289 doi: 10.1007/BF01931289
[59]	Q. Zhang, L. Liu, C. Zhang, Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays, Appl. Anal., 101 (2022), 1911–1932. https://doi.org/10.1080/00036811.2020.1789600 doi: 10.1080/00036811.2020.1789600
[60]	Z. Y. Zhou, H. X. Zhang, X. H. Yang, The compact difference scheme for the fourth-order nonlocal evolution equation with a weakly singular kernel, Math. Method Appl. Sci., 46 (2023), 5422–5447. https://doi.org/10.1002/mma.8842 doi: 10.1002/mma.8842
[61]	J. W. Wang, X. X. Jiang, X. H. Yang, H. X. Zhang, A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers' type nonlinearity, J. Appl. Math. Comput., 70 (2024), 489–511. https://doi.org/10.1007/s12190-023-01975-4 doi: 10.1007/s12190-023-01975-4
[62]	X. Mao, Q. Zhang, D. Xu, Y. Xu, Double reduction order method based conservative compact schemes for the Rosenau equation, Appl. Numer. Math., 197 (2024), 15–45. https://doi.org/10.1016/j.apnum.2023.11.001 doi: 10.1016/j.apnum.2023.11.001
[63]	W. Wang, H. X. Zhang, Z. Y. Zhou, X. H. Yang, A fast compact finite difference scheme for the fourth-order diffusion-wave equation, Int. J. Comput. Math., (2024), 1–24. https://doi.org/10.1080/00207160.2024.2323985 doi: 10.1080/00207160.2024.2323985
[64]	J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, in Texts in Applied Mathematics, Springer, Berlin, 1995.
[65]	J. Zhao, X. Jiang, Y. Xu, Generalized Adams method for solving fractional delay differential equations, Math. Comput. Simulat., 180 (2021), 401–419. https://doi.org/10.1016/j.matcom.2020.09.006 doi: 10.1016/j.matcom.2020.09.006
[66]	F. R. Wang, X. H. Yang, H. X. Zhang, L. J. Wu, A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel, Math. Comput. Simulat., 199 (2022), 38–59. https://doi.org/10.1016/j.matcom.2022.03.004 doi: 10.1016/j.matcom.2022.03.004
[67]	X. H. Yang, H. X. Zhang, The uniform $l^1$ long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
[68]	C. J. Li, H. X. Zhang, X. H. Yang, A new $\alpha$-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation, Commun. Anal. Mech., 16 (2024), 147–168. https://doi.org/10.3934/cam.2024007 doi: 10.3934/cam.2024007
[69]	W. Xiao, X. H. Yang, Z. Z. Zhou, Pointwise-in-time $\alpha$-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients, Commun. Anal. Mech., 16 (2024), 53–70. https://doi.org/10.3934/cam.2024003 doi: 10.3934/cam.2024003

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)