×

Numerical solution of the heat conduction problem with memory. (English) Zbl 1524.65984

Summary: It is necessary to use more general models than the classical Fourier heat conduction law to describe small-scale thermal conduction processes. The effects of heat flow memory and heat capacity memory (internal energy) in solids are considered in first-order integrodifferential evolutionary equations with difference-type kernels. The main difficulties in applying such nonlocal in-time mathematical models are associated with the need to work with a solution throughout the entire history of the process. The paper develops an approach to transforming a nonlocal problem into a computationally simpler local problem for a system of first-order evolution equations. Such a transition is applicable for heat conduction problems with memory if the relaxation functions of the heat flux and heat capacity are represented as a sum of exponentials. The correctness of the auxiliary linear problem is ensured by the obtained estimates of the stability of the solution concerning the initial data and the right-hand side in the corresponding Hilbert spaces. The study’s main result is to prove the unconditional stability of the proposed two-level scheme with weights for the evolutionary system of equations for modeling heat conduction in solid media with memory. In this case, finding an approximate solution on a new level in time is not more complicated than the classical heat equation. The numerical solution of a model one-dimensional in space heat conduction problem with memory effects is presented.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
45K05 Integro-partial differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Joseph, D. D.; Preziosi, L., Heat waves, Rev. Mod. Phys., 61, 1, 41 (1989) · Zbl 1129.80300
[2] Straughan, B., Heat Waves (2011), Springer Science & Business Media · Zbl 1250.80001
[3] Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal., 31, 2, 113-126 (1968) · Zbl 0164.12901
[4] Nunziato, J. W., On heat conduction in materials with memory, Q. Appl. Math., 29, 2, 187-204 (1971) · Zbl 0227.73011
[5] Li, S.-N.; Cao, B.-Y., Fractional-order heat conduction models from generalized Boltzmann transport equation, Philos. Trans. R. Soc. A, 378, 2172, Article 20190280 pp. (2020) · Zbl 1462.82062
[6] Li, S.-N.; Cao, B.-Y., Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity, Int. J. Heat Mass Transf., 137, 84-89 (2019)
[7] Li, S. N.; Cao, B. Y., Generalized Boltzmann transport theory for relaxational heat conduction, Int. J. Heat Mass Transf., 173, Article 121225 pp. (2021)
[8] Xu, M.; Li, X., The modeling of nanoscale heat conduction by Boltzmann transport equation, Int. J. Heat Mass Transf., 55, 7-8, 1905-1910 (2012)
[9] Chen, C.; Shih, T., Finite Element Methods for Integrodifferential Equations (1998), World Scientific: World Scientific Singapore · Zbl 0909.65142
[10] Linz, P., Analytical and Numerical Methods for Volterra Equations (1985), SIAM · Zbl 0566.65094
[11] Braess, D., Nonlinear Approximation Theory (1986), Springer: Springer Berlin, Heidelberg · Zbl 0656.41001
[12] Vabishchevich, P. N., Numerical solution of the Cauchy problem for Volterra integrodifferential equations with difference kernels, Appl. Numer. Math., 174, 177-190 (2022) · Zbl 1485.65137
[13] Vabishchevich, P. N., Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory (2021), pp. 1-13
[14] Cattaneo, C., Sulla conduzione del calore, Atti Semin. Mat. Fis. Univ. Modena, 3, 83-101 (1948) · Zbl 0035.26203
[15] Bubnov, V. A., Wave concepts in the theory of heat, Int. J. Heat Mass Transf., 19, 2, 175-184 (1976) · Zbl 0316.35045
[16] Davis, P. L., On the hyperbolicity of the equations of the linear theory of heat conduction for materials with memory, SIAM J. Appl. Math., 30, 1, 75-80 (1976) · Zbl 0341.45027
[17] Davis, P. L., On the linear theory of heat conduction for materials with memory, SIAM J. Math. Anal., 9, 1, 49-53 (1978) · Zbl 0391.45011
[18] Knabner, P.; Angermann, L., Numerical Methods for Elliptic and Parabolic Partial Differential Equations (2003), Springer: Springer New York · Zbl 1034.65086
[19] Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer-Verlag: Springer-Verlag Berlin · Zbl 0803.65088
[20] LeVeque, R. J., Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-State and Time-Dependent Problems (2007), Society for Industrial Mathematics · Zbl 1127.65080
[21] Samarskii, A. A., The Theory of Difference Schemes (2001), Marcel Dekker: Marcel Dekker New York · Zbl 0971.65076
[22] Shen, B.; Zhang, P., Notable physical anomalies manifested in non-Fourier heat conduction under the dual-phase-lag model, Int. J. Heat Mass Transf., 51, 7-8, 1713-1727 (2008) · Zbl 1140.80368
[23] Hu, R. F.; Cao, B. Y., Study on thermal wave based on the thermal mass theory, Sci. China Ser. E: Technol. Sci., 52, 6, 1786-1792 (2009) · Zbl 1177.80026
[24] Zhang, M.-K.; Cao, B.-Y.; Guo, Y.-C., Numerical studies on dispersion of thermal waves, Int. J. Heat Mass Transf., 67, 1072-1082 (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.