A FINITE VOLUME METHOD PRESERVING MAXIMUM PRINCIPLE FOR THE CONJUGATE HEAT TRANSFER PROBLEMS WITH GENERAL INTERFACE CONDITIONS

doi:10.4208/jcm.2107-m2020-0266

Journal of Computational Mathematics ›› 2023, Vol. 41 ›› Issue (3): 345-369.DOI: 10.4208/jcm.2107-m2020-0266

A FINITE VOLUME METHOD PRESERVING MAXIMUM PRINCIPLE FOR THE CONJUGATE HEAT TRANSFER PROBLEMS WITH GENERAL INTERFACE CONDITIONS

Huifang Zhou^1,2, Zhiqiang Sheng^3,4, Guangwei Yuan³

1. School of Mathematics, Jilin University, Changchun 130012, China;
2. The Graduate School of China Academy of Engineering Physics, Beijing 100088, China;
3. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
4. HEDPS, Center for Applied Physics and Technology, and College of Engineering, Peking University, Beijing 100871, China

Received:2020-10-04 Revised:2021-03-05 Published:2023-03-14
Contact: Guangwei Yuan, Email:yuan_guangwei@iapcm.ac.cn
Supported by:
This work is partially supported by the National Natural Science Foundation of China (11971069,12071045), and the Foundation of CAEP (CX20210042), and Science Challenge Project (No. TZ2016002).

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Huifang Zhou, Zhiqiang Sheng, Guangwei Yuan. A FINITE VOLUME METHOD PRESERVING MAXIMUM PRINCIPLE FOR THE CONJUGATE HEAT TRANSFER PROBLEMS WITH GENERAL INTERFACE CONDITIONS[J]. Journal of Computational Mathematics, 2023, 41(3): 345-369.

In this paper, we present a unified finite volume method preserving discrete maximum principle (DMP) for the conjugate heat transfer problems with general interface conditions. We prove the existence of the numerical solution and the DMP-preserving property. Numerical experiments show that the nonlinear iteration numbers of the scheme in [24] increase rapidly when the interfacial coefficients decrease to zero. In contrast, the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero, which reveals that the unified scheme is more robust than the scheme in [24]. The accuracy and DMP-preserving property of the scheme are also verified in the numerical experiments.

Key words: Conjugate heat transfer problems, General interface conditions, Finite volume scheme, Discrete maximum principle

CLC Number:

65M08
35K59

[1] A. Cangiani, E.H. Georgoulis, and Y. A. Sabawi, Adaptive discontinuous Galerkin methods for elliptic interface problems, Math. Comput., 87(2018), 2675-2707.
[2] F. Cao, Z. Sheng, and G. Yuan, Monotone finite volume schemes for diffusion equation with imperfect interface on distorted meshes, J. Sci. Comput., 76(2018), 1055-1077.
[3] T. Chernogorova, R.E. Ewing, O. Iliev, and R. Lazarov, Numer. Treat. Multiph. Flows Porous Media, Springer Berlin Heidelberg, 2000.
[4] R. Costa, J.M. Nóbrega, S. Clain, and G. J. Machado, Very high-order accurate polygonal mesh finite volume scheme for conjugate heat transfer problems with curved interfaces and imperfect contacts, Comput. Methods Appl. Mech. Eng., 357(2019), 112560.
[5] X. He, T. Lin, and Y. Lin, A selective immersed discontinuous Galerkin method for elliptic interface problems, Math. Methods Appl. Sci., 37(2014), 983-1002.
[6] H. Huang, J. Li, and J. Yan, High order symmetric direct discontinuous Galerkin method for elliptic interface problems with fitted mesh, J. Comput. Phys., 409(2020), 109301.
[7] A. Jain, R. Kanapady, and K. Tamma, Local discontinuous Galerkin method for parabolic problems involving imperfect contact surfaces, Collection of Technical Papers-44th AIAA Aerospace Sciences Meeting, 10(2006), 7076-7086.
[8] D. Jia, Z. Sheng, and G. Yuan, An extremum-preserving iterative procedure for the imperfect interface problem, Commun. Comput. Phys., 25(2019), 1-18.
[9] B.S. Jovanovic, L.D. Ivanovic, and E.E. Suli, Convergence of finite-difference schemes for elliptic equations with variable coefficients, IMA J. Numer. Anal., 7(1987), 301-305.
[10] J. Kačur and R. Van Keer, Numerical method for a class of parabolic problems in composite media, Numer. Methods Partial Differ. Equ., 9(1993), 711-731.
[11] Z. Li and K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23(2001), 339-361.
[12] T. Lin, Y. Lin, and X. Zhang, Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53(2015), 1121-1144.
[13] T. Lin, Q. Yang, and X. Zhang, A priori error estimates for some discontinuous Galerkin immersed finite element methods, J. Sci. Comput., 65(2015), 875-894.
[14] K. Lipnikov, M. Shashkov, D. Svyatskiy, and Y. Vassilevski, Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, J. Comput. Phys., 227(2007), 492-512.
[15] A.W. Pratt, Heat transmission in buildings, Wiley, Chichester, 1981.
[16] P. Rodrigo, A. Rocha, and M. E. Cruz, Computation of the effective conductivity of unidirectional fibrous composites with an interfacial thermal resistance, Numer. Heat Transf. Part A Appl., 39(2001), 179-203.
[17] Z. Sheng and G. Yuan, A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., 30(2008), 1341-1361.
[18] Z. Sheng and G. Yuan, The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes, J. Comput. Phys., 230(2011), 2588-2604.
[19] L. Wang, S. Hou, and L. Shi, A weak formulation for solving the elliptic interface problems with imperfect contact, Adv. Appl. Math. Mech., 9(2017), 1189-1205.
[20] J. Weisz, On an iterative method for the solution of discretized elliptic problems with imperfect contact condition, J. Comput. Appl. Math., 72(1996), 319-333.
[21] J. Xu, F. Zhao, Z. Sheng, and G. Yuan, A nonlinear finite volume scheme preserving maximum principle for diffusion equations, Commun. Comput. Phys., 29(2021), 747-766.
[22] Q. Yang and X. Zhang, Discontinuous Galerkin immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 299(2016), 127-139.
[23] G. Yuan and Z. Sheng, Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phys., 227(2008), 6288-6312.
[24] H. Zhou, Z. Sheng, and G. Yuan, A finite volume method preserving maximum principle for the diffusion equations with imperfect interface, Appl. Numer. Math., 158(2020), 314-335.
[25] Q. Zhuang and R. Guo, High degree discontinuous Petrov-Galerkin immersed finite element methods using fictitious elements for elliptic interface problems, J. Comput. Appl. Math., 362(2019), 560-573.

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A FINITE VOLUME METHOD PRESERVING MAXIMUM PRINCIPLE FOR THE CONJUGATE HEAT TRANSFER PROBLEMS WITH GENERAL INTERFACE CONDITIONS

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