Optimized Schwarz waveform relaxation for heterogeneous Cattaneo-Vernotte non-Fourier heat transfer

Feng Hu; Yingxiang Xu; Feng Hu; Yingxiang Xu

doi:10.3934/math.2025338

AIMS Mathematics

2025, Volume 10, Issue 3: 7370-7395. doi: 10.3934/math.2025338

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Optimized Schwarz waveform relaxation for heterogeneous Cattaneo-Vernotte non-Fourier heat transfer

Feng Hu ,
Yingxiang Xu ^,

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, People's Republic of China

Received: 08 December 2024 Revised: 24 February 2025 Accepted: 19 March 2025 Published: 31 March 2025
MSC : 35Q79, 65M55

The non-Fourier heat transfer in heterogeneous media is crucial for material science and biomedical engineering. The optimized Schwarz waveform relaxation (OSWR) method is an efficient approach for solving such problems due to its divide-and-conquer strategy. Despite the wave-type nature of non-Fourier heat transfer, the short phase-lag time leads to more parabolic-like behavior. To address this, in the OSWR method, we employed Robin boundary conditions to transmit information along the interface. Using Fourier analysis, we derived and rigorously optimized the convergence factors of the OSWR algorithm with scaled Robin and Robin-Robin transmission conditions. The resulting optimized transmission parameters were provided in explicit form for direct application in the OSWR algorithm, along with corresponding convergence factor estimates. Interestingly, the results show that a larger heterogeneity contrast actually accelerates the convergence, rather than deteriorating it. Furthermore, the OSWR algorithm with the Robin-Robin condition exhibits mesh-independent convergence asymptotically. However, the presence of the phase-lag time is found to slow down the convergence of the OSWR algorithm. These theoretical findings were validated through numerical experiments.
- Cattaneo-Vernotte equation,
- optimized Schwarz waveform relaxation,
- heterogeneous media,
- Robin transmission condition,
- non-Fourier heat transfer
Citation: Feng Hu, Yingxiang Xu. Optimized Schwarz waveform relaxation for heterogeneous Cattaneo-Vernotte non-Fourier heat transfer[J]. AIMS Mathematics, 2025, 10(3): 7370-7395. doi: 10.3934/math.2025338

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Abstract

The non-Fourier heat transfer in heterogeneous media is crucial for material science and biomedical engineering. The optimized Schwarz waveform relaxation (OSWR) method is an efficient approach for solving such problems due to its divide-and-conquer strategy. Despite the wave-type nature of non-Fourier heat transfer, the short phase-lag time leads to more parabolic-like behavior. To address this, in the OSWR method, we employed Robin boundary conditions to transmit information along the interface. Using Fourier analysis, we derived and rigorously optimized the convergence factors of the OSWR algorithm with scaled Robin and Robin-Robin transmission conditions. The resulting optimized transmission parameters were provided in explicit form for direct application in the OSWR algorithm, along with corresponding convergence factor estimates. Interestingly, the results show that a larger heterogeneity contrast actually accelerates the convergence, rather than deteriorating it. Furthermore, the OSWR algorithm with the Robin-Robin condition exhibits mesh-independent convergence asymptotically. However, the presence of the phase-lag time is found to slow down the convergence of the OSWR algorithm. These theoretical findings were validated through numerical experiments.

References

[1]	A. Narasimhan, S. Sadasivam, Non-Fourier bio heat transfer modelling of thermal damage during retinal laser irradiation, Int. J. Heat Mass Tran., 60 (2013), 591–597. https://doi.org/10.1016/j.ijheatmasstransfer.2013.01.010 doi: 10.1016/j.ijheatmasstransfer.2013.01.010
[2]	J. Zhou, Y. Zhang, J. Chen, Non-Fourier heat conduction effect on laser-induced thermal damage in biological tissues, Numer. Heat Tr. A-Appl., 54 (2008), 1–19. https://doi.org/10.1080/10407780802025911 doi: 10.1080/10407780802025911
[3]	T. Xue, X. Zhang, K. K. Tamma, Investigation of thermal interfacial problems involving non-locality in space and time, Int. Commun. Heat Mass, 99 (2018), 37–42. https://doi.org/10.1016/j.icheatmasstransfer.2018.10.008 doi: 10.1016/j.icheatmasstransfer.2018.10.008
[4]	Z. Y. Guo, Y. S. Xu, Non-Fourier heat conduction in IC chip, J. Electron. Packag. Sep., 117 (1995), 174–177. https://doi.org/10.1115/1.2792088 doi: 10.1115/1.2792088
[5]	H. D. Wang, B. Y. Cao, Z. Y. Guo, Non-Fourier heat conduction in carbon nanotubes, J. Heat Transfer, 134 (2012), 051004. https://doi.org/10.1115/1.4005634 doi: 10.1115/1.4005634
[6]	C. Cattaneo, A form of heat-conduction equations which eliminates the paradox of instantaneous propagation, Comptes Rendus, 247 (1958), 431.
[7]	P. Vernotte, Some possible complications in the phenomena of thermal conduction, Compte Rendus, 252 (1961), 2190–2191.
[8]	M. Ozisik, D. Y. Tzou, On the wave theory in heat conduction, J. Heat Transfer, 116 (1994), 526–535. https://doi.org/10.1115/1.2910903 doi: 10.1115/1.2910903
[9]	J. Ordonez-Miranda, J. Alvarado-Gil, Thermal wave oscillations and thermal relaxation time determination in a hyperbolic heat transport model, Int. J. Therm. Sci., 48 (2009), 2053–2062. https://doi.org/10.1016/j.ijthermalsci.2009.03.008 doi: 10.1016/j.ijthermalsci.2009.03.008
[10]	A. Vedavarz, S. Kumar, M. K. Moallemi, Significance of non-Fourier heat waves in conduction, J. Heat Transfer, 116 (1994), 221–224. https://doi.org/10.1115/1.2910859 doi: 10.1115/1.2910859
[11]	D. Y. Tzou, J. C. Dowell, Computational techniques for microscale heat transfer, In: Handbook of Numerical Heat Transfer, 2000. https://doi.org/10.1002/9780470172599.ch20
[12]	K. C. Liu, P. J. Cheng, Finite propagation of heat transfer in a multilayer tissue, J. Thermophys. Heat Tr., 22 (2008), 775–782. https://doi.org/10.2514/1.37267 doi: 10.2514/1.37267
[13]	C. Barman, P. Rath, A. Bhattacharya, A non-Fourier bioheat transfer model for cryosurgery of tumor tissue with minimum collateral damage, Comput. Meth. Prog. Bio., 200 (2021), 105857. https://doi.org/10.1016/j.cmpb.2020.105857 doi: 10.1016/j.cmpb.2020.105857
[14]	P. Namakshenas, A. Mojra, Microstructure-based non-Fourier heat transfer modeling of HIFU treatment for thyroid cancer, Comput. Meth. Prog. Bio., 197 (2020), 105698. https://doi.org/10.1016/j.cmpb.2020.105698 doi: 10.1016/j.cmpb.2020.105698
[15]	Z. S. Deng, J. Liu, Non-Fourier heat conduction effect on prediction of temperature transients and thermal stress in skin cryopreservation, J. Therm. Stresses, 26 (2003), 779–798. https://doi.org/10.1080/01495730390219377 doi: 10.1080/01495730390219377
[16]	A. K. Kheibari, M. Jafari, M. B. Nazari, Propagation of heat wave in composite cylinder using Cattaneo-Vernotte theory, Int. J. Heat Mass Transfer, 160 (2020), 120208. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120208 doi: 10.1016/j.ijheatmasstransfer.2020.120208
[17]	A. Fehér, R. Kovacs, Novel evaluation method for non-Fourier effects in heat pulse experiments, 2021. https://doi.org/10.48550/arXiv.2101.01123
[18]	A. Fehér, D. Markovics, T. Fodor, R. Kovacs, Size effects and non-Fourier thermal behaviour in rocks, ISRM EUROCK, 2020.
[19]	P. Duhamel, A new finite integral transform pair for hyperbolic conduction problems in heterogeneous media, Int. J. Heat Mass Transfer, 44 (2001), 3307-3320. https://doi.org/10.1016/S0017-9310(00)00360-4 doi: 10.1016/S0017-9310(00)00360-4
[20]	S. Singh, Z. Li, A high order compact scheme for a thermal wave model of bio-heat transfer with an interface, Numer. Math. Theory Me., 11 (2018), 321–337. https://doi.org/10.4208/nmtma.OA-2017-0048 doi: 10.4208/nmtma.OA-2017-0048
[21]	M. Asif, F. Bilal, R. Bilal, N. Haider, S. A. M. Abdelmohsenc, S. M. Eldind, An efficient algorithm for the numerical solution of telegraph interface model with discontinuous coefficients via Haar wavelets, Alex. Eng. J., 72 (2023), 275–285. https://doi.org/10.1016/j.aej.2023.03.074 doi: 10.1016/j.aej.2023.03.074
[22]	T. Ahmod, J. Dutta, Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension, Proc. Math. Sci., 132 (2022), 6. https://doi.org/10.1007/s12044-021-00646-3 doi: 10.1007/s12044-021-00646-3
[23]	B. Deka, J. Dutta, Finite element methods for non-Fourier thermal wave model of bio heat transfer with an interface, J. Appl. Math. Comput., 62 (2020), 701–724. https://doi.org/10.1007/s12190-019-01304-8 doi: 10.1007/s12190-019-01304-8
[24]	B. Deka, J. Dutta, Convergence of finite element methods for hyperbolic heat conduction model with an interface, Comput. Math. Appl., 79 (2020), 3139–3159. https://doi.org/10.1016/j.camwa.2020.01.013 doi: 10.1016/j.camwa.2020.01.013
[25]	J. Dutta, B. Deka, Optimal a priori error estimates for the finite element approximation of dual-phase-lag bio heat model in heterogeneous medium, J. Sci. Comput., 87 (2021), 58. https://doi.org/10.1007/s10915-021-01460-9 doi: 10.1007/s10915-021-01460-9
[26]	S. Han, Finite volume solution of 2-D hyperbolic conduction in a heterogeneous medium, Numer. Heat Tr. A-Appl., 70 (2016), 723–737. https://doi.org/10.1080/10407782.2016.1193347 doi: 10.1080/10407782.2016.1193347
[27]	H. Sauerland, T. P. Fries, The stable XFEM for two-phase flows, Comput. Fluids, 87 (2013), 41–49. https://doi.org/10.1016/j.compfluid.2012.10.017 doi: 10.1016/j.compfluid.2012.10.017
[28]	B. Ayuso de Dios, M. Holst, Y. Zhu, L. Zikatanov, Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients, Math. Comp., 83 (2014), 1083–1120. https://doi.org/10.1090/S0025-5718-2013-02760-3 doi: 10.1090/S0025-5718-2013-02760-3
[29]	M. J. Gander, Optimized Schwarz methods, SIAM J. Numer. Anal., 44 (2006), 699–731. https://doi.org/10.1137/S0036142903425409 doi: 10.1137/S0036142903425409
[30]	Y. Maday, F. Magoules, Optimized Schwarz methods without overlap for highly heterogeneous media, Comput. Method. Appl. M., 196 (2007), 1541–1553. https://doi.org/10.1016/j.cma.2005.05.059 doi: 10.1016/j.cma.2005.05.059
[31]	M. J. Gander, O. Dnbois, Optimized Schwarz methods for a diffusion problem with discontinuous coefficient, Numer. Algor., 69 (2015), 109–144. https://doi.org/10.1007/s11075-014-9884-2 doi: 10.1007/s11075-014-9884-2
[32]	M. J. Gander, T. Vanzan, Heterogeneous optimized Schwarz methods for second order elliptic PDEs, SIAM J. Sci. Comput., 41 (2019), A2329–A2354. https://doi.org/10.1137/18M122114X doi: 10.1137/18M122114X
[33]	M. J. Gander, S. B. Lunowa, Ch. Rohde, Non-overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations, SIAM J. Sci. Comput., 45 (2023), A49–A73. https://doi.org/10.1137/21M1415005 doi: 10.1137/21M1415005
[34]	M. J. Gander, L. Halpern, M. Kern, A Schwarz waveform relaxation method for advection-diffusion-reaction problems with discontinuous coefficients and non-matching grids, In: Domain Decomposition Methods in Science and Engineering, Berlin: Springer, 2007,283–290. https://doi.org/10.1007/978-3-540-34469-8_33
[35]	L. Halpern, C. Japhet, Discontinuous Galerkin and nonconforming in time optimized Schwarz waveform relaxation for heterogeneous problems, In: Domain Decomposition Methods in Science and Engineering XVII, Berlin: Springer, 2008,211–219. https://doi.org/10.1007/978-3-540-75199-1_23
[36]	L. Halpern, C. Japhet, J. Szeftel, Optimized Schwarz waveform relaxation and discontinuous Galerkin time stepping for heterogeneous problems, SIAM J. Numer. Anal., 50 (2012), 2588–2611. https://doi.org/10.1137/120865033 doi: 10.1137/120865033
[37]	M. D. Al-Khaleel, M. J. Gander, and P. M. Kumbhar, Optimized Schwarz waveform relaxation methods for the telegrapher equation, SIAM J. Sci. Comput., 46 (2024), A3528–A3551. https://doi.org/10.1137/24M1642962 doi: 10.1137/24M1642962
[38]	M. J. Gander, J. Lin, S. L. Wu, X. Yue, T. Zhou, Parareal: parallel-in-time algorithms based on the diagonalization technique, 2020. https://doi.org/10.48550/arXiv.2005.09158
[39]	M. J. Gander, S. Vandewalle, Analysis of the Parareal time-parallel time integration method, SIAM J. Sci. Comput., 29 (2007), 556–578. https://doi.org/10.1137/05064607X doi: 10.1137/05064607X
[40]	G. Ciaramella, M. J. Gander, I. Mazzieri, Unmapped tent pitching schemes by waveform relaxation, In: Decomposition Methods in Science and Engineering XXVII, Cham: Springer, 2024,455–462. https://doi.org/10.1007/978-3-031-50769-4_54
[41]	M. J. Gander, L. Halpern, Optimized Schwarz waveform relaxation methods for advection-reaction-diffusion problems, SIAM J. Numer. Anal., 45 (2007), 666–697. https://doi.org/10.1137/050642137 doi: 10.1137/050642137
[42]	D. Bennequin, M. J. Gander, L. Gouarin, L. Halpern, Optimized Schwarz waveform relaxation for advection-reaction-diffusion equations in two dimensions, Numer. Math., 134 (2016), 513–567. https://doi.org/10.1007/s00211-015-0784-8 doi: 10.1007/s00211-015-0784-8
[43]	V. Martin, An optimized Schwarz waveform relaxation method for the unsteady convection-diffusion equation in two dimensions, Appl. Numer. Math., 52 (2005), 401–428. https://doi.org/10.1016/j.apnum.2004.08.022 doi: 10.1016/j.apnum.2004.08.022
[44]	Y. Xu, The influence of domain truncation on the performance of optimized Schwarz methods, Electron. T. Numer. Ana., 49 (2018), 182–209. https://doi.org/10.1553/etna_vol49s182 doi: 10.1553/etna_vol49s182
[45]	W. B. Dong, H. S. Tang, Convergence analysis of Schwarz waveform relaxation method to compute coupled advection-diffusion-reaction equations, Math. Comput. Simul., 218 (2024), 462–481. https://doi.org/10.1016/j.matcom.2023.11.026 doi: 10.1016/j.matcom.2023.11.026
[46]	M. J. Gander, V. Martin, Why Fourier mode analysis in time is different when studying Schwarz waveform relaxation, J. Comput. Phys., 491 (2023), 112316. https://doi.org/10.1016/j.jcp.2023.112316 doi: 10.1016/j.jcp.2023.112316

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