Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method

Majchrzak, Ewa; Mochnacki, B.

doi:10.17512/jamcm.2016.3.09

Cited by 23 publications

(21 citation statements)

References 10 publications

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“…The second profile corresponds to ∆t = 0.01 ps, while the last (significantly different from the previous) has been computed for ∆t = 0.05 ps. The time interval ∆t = 0.01 ps is longer than the critical one for the FDM explicit scheme [10]. In spite of this, the solution is of a good accuracy.…”

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confidence: 98%

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

Majchrzak

Mochnacki

2017

J. Appl. Math. Comput. Mech.

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Abstract. 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.MSC 2010: 80M20, 80A20

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confidence: 98%

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

Majchrzak

Mochnacki

2017

J. Appl. Math. Comput. Mech.

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“…As one knows, a stability condition should be formulated for these types of algorithms. The considerations concerning the formulation of the condition discussed are presented in [2] and [3]. The implicit scheme of the FDM was also the topic of our research work, here the papers [4,5] should be mentioned.…”

Section: Microscale Heat Transfer 1d and 2d Problemsmentioning

confidence: 99%

Application of numerical methods for solving the non-fourier equations. A review of our own and collaborators’ works

Majchrzak¹,

Mochnacki²

2018

J. Appl. Math. Comput. Mech.

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Abstract. Thermal processes occuring in the solid bodies are, as a rule, described by the well-known Fourier equation (or the system of these equations) supplemented by the appropriate boundary and initial conditions. Such a mathematical model is sufficiently exact to describe the heat transfer processes in the macro scale for the typical materials. It turned out that the energy equation based on the Fourier law has the limitations and it should not be used in the case of the microscale heat transfer and also in the case of materials with a special inner structure (e.g. biological tissue). The better approximation of the real thermal processes assure the modifications of the energy equation, in particular the models in which the so-called lag times are introduced. The article presented is devoted to the numerical aspects of solving these types of equations (in the scope of the microscale heat transfer). The results published by the other authors can be found in the references posted in the works cited below. MSC 2010: 35L10, 65M06Keywords: heat transfer, non-Fourier heat diffusion models, numerical methods, computational mechanics Governing equationsIn the well-known Fourier law, the relationship between the heat flux and the temperature gradient for the heat conduction process is of the formwhere q is a heat flux vector, λ is a thermal conductivity, T, X, t denote the temperature, spatial co-ordinates and time.

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“…After solving the problem (23), (24), (15), the ordinary differential equation (17) with the initial condition (c.f. formula (16)…”

Section: Modifications Of Dplementioning

confidence: 99%

First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method

Majchrzak¹,

Mochnacki

2018

MATEC Web Conf.

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In the paper the different variants of the dual phase lag equation (DPLE) are considered. As one knows, the mathematical form of DPLE results from the generalization of the Fourier law in which two delay times are introduced, namely the relaxation time τq and the thermalization one τT. Depending on the order of development of the left and right hand sides of the generalized Fourier law into the Taylor series one can obtain the different forms of the DPLE. It is also possible to consider the others forms of equation discussed resulting from the introduction of the new variable or variables (substitution). In the paper a thin metal film subjected to a laser pulse is considered (the 1D problem). Theoretical considerations are illustrated by the examples of numerical computations. The discussion of the results obtained is also presented.

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Dual-phase lag equation. Stability conditions of a numerical algorithm based on the explicit scheme of the finite difference method

Cited by 23 publications

References 10 publications

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

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

Application of numerical methods for solving the non-fourier equations. A review of our own and collaborators’ works

First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method

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