Implicit scheme of the finite difference method for 1D dual-phase lag equation

• Autorzy: Majchrzak E. , Mochnacki B.

Identyfikatory

DOI

10.17512/jamcm.2017.3.04

• Języki publikacji: EN

Abstrakty

EN

The 1D dual-phase lag equation (DPLE) is solved using the implicit FDM scheme. The dual phase lag equation is the hyperbolic PDE and contains a second order time derivative and higher order mixed derivative in both time and space. The DPLE results from the generalization of the well known Fourier law in which the delay times are taken into account. So, in the equation discussed, two positive parameters appear. They correspond to the relaxation time τ_q and the thermalization time τ _T. The DPLE finds, among others, the application as the mathematical description of the thermal processes proceeding in the micro-scale. In the paper, the numerical solution of DPLE based on the implicit scheme of the FDM is presented. The authors show that a such an approach in the case of DPLE leads to the unconditionally stable differential scheme.

Słowa kluczowe

EN

micro-scale heat conduction; dual-phase lag equation; finite difference method; stability of FDM implicit scheme

PL

równanie z dwoma czasami opóźnień; metoda różnic skończonych; DPLE; schemat FDM

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2017
• Tom: Vol. 16, nr 3
• Strony: 37--46
• Opis fizyczny: Bibliogr. 12 poz., rys.

Twórcy

autor

Majchrzak E.

• Silesian University of Technology Gliwice, Poland

autor

Mochnacki B.

• University of Occupational Safety Management in Katowice Katowice, Poland

Bibliografia

• [1] Tzou D.Y., Macro- to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons, Ltd 2015.
• [2] Zhang Z.M., Nano/Microscale Heat Transfer, McGraw-Hill, New York 2007.
• [3] Escobar R.A., Ghai S.S., Jhon M.S., Amon C.H., Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronic cooling, International Journal of Heat and Mass Transfer 2006, 49, 97-107.
• [4] Majchrzak E., Mochnacki B., Suchy J.S., Numerical simulation of thermal processes proceeding in a multi-layered film subjected to ultrafast laser heating, Journal of Theoretical and Applied Mechanics 2009, 47, 2, 383-396.
• [5] Al-Nimr M.A., Heat transfer mechanism during short duration laser heating in thin metal films, International Journal of Thermophysics 1997, 18(5), 1257-1268.
• [6] Majchrzak E., Mochnacki B., Sensitivity analysis of transient temperature field in microdomains with respect to the dual phase lag model parameters, International Journal for Multiscale Computational Engineering 2014, 12(1), 65-77.
• [7] McDonough J.M., Kunadian I., Kumar R.R., Yang T., An alternative discretization and solution procedure for the dual phase-lag equation, Journal of Computational Physics 2006, 219, 163-171.
• [8] Young D.M., Iterative Solution of Large Linear Systems, Academic Press, New York 1971.
• [9] Ciesielski M., Analytical solution of the dual phase lag equation describing the laser heating of thin metal film, Journal of Applied Mathematics and Computational Mechanics, 2017, 16(1), 33-40.
• [10] Majchrzak E., Mochnacki B., Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method, Journal of Applied Mathematics and Computational Mechanics 2016, 15(3), 89-96.
• [11] Castro M.A., Martin J.A., Rodriguez F., Unconditional stability of a numerical method for the dual-phase-lag equation, Mathematical Problems in Engineering 2017, 5 pages.
• [12] Kaba I.K., Dai W., A stable three-level finite difference scheme for solving the parabolic twostep model in a 3D micro-sphere heated by ultrashort-pulsed lasers, Journal of Computational and Applied Mathematics 2005, 181, 125-147.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).