An unconditionally stable finite difference scheme for solving a 3D heat transport equation in a sub-microscale thin film. (English) Zbl 1005.65084

Dai, Weizhong; Nassar, Raja

J. Comput. Appl. Math. 145, No. 1, 247-260 (2002).

The authors develop a finite difference scheme of the Crank-Nicolson type by introducing an intermediate function to the heat transport equation. The scheme contains two levels in time. It is shown by the discrete energy method that this scheme is unconditionally stable with respect to the initial values. The numerical procedure is employed to obtain the temperature rise in a gold sub-microscale thin film.

Reviewer: Prabhat Kumar Mahanti (New Brunswick)

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05	Heat equation
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
80A20	Heat and mass transfer, heat flow (MSC2010)
80M10	Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer

Keywords:

discrete energy method; heat transport equation; gold sub-microscale thin film; Crank-Nicolson finite difference method; stability

References:

[1]	R.J. Chiffell, On the wave behavior and rate effect of thermal and thermomechanical waves, M.S. Thesis, University of New Mexico, Albuquerque, NM, 1994.
[2]	Dai, W.; Nassar, R., A finite difference scheme for solving the heat transport equation at the microscale, Numer. methods partial differential equations, 15, 697-708, (1999) · Zbl 0945.65096
[3]	Dai, W.; Nassar, R., A finite difference scheme for solving a three-dimensional heat transport equation in a thin film with microscale thickness, Internat. J. numer. methods engng., 50, 1165-1180, (2001)
[4]	Dai, W.; Nassar, R., A domain decomposition method for solving thin film elliptic interface problems with variable coefficients, Int. J. numer. methods engng., 46, 747-756, (1999) · Zbl 1073.74640
[5]	Dai, W.; Nassar, R., A hybrid finite element-finite difference for thermal analysis in X-ray lithography, Internat. J. numer. methods heat fluid flow, 9, 660-676, (1999) · Zbl 0953.80002
[6]	Dai, W.; Nassar, R., A preconditioned Richardson numerical method for thermal analysis in X-ray lithography with cylindrical geometry, Numer. heat transfer part A, 34, 599-616, (1998)
[7]	Dai, W.; Nassar, R., A three-dimensional numerical method for thermal analysis in X-ray lithography, Internat. J. numer. methods heat fluid flow, 8, 409-423, (1998) · Zbl 0935.78017
[8]	Incropera, E.P.; Dewitt, D.P., Fundamentals of heat and mass transfer, (1990), Wiley New York
[9]	Joseph, D.D.; Preziosi, L., Heat waves, Rev. modern phys., 61, 41-73, (1989) · Zbl 1129.80300
[10]	Joshi, A.A.; Majumdar, A., Transient ballistic and diffusive phonon heat transport in thin films, J. appl. phys., 74, 31-39, (1993)
[11]	Lees, M., Alternating direction and semi-explicit difference methods for parabolic partial differential equations, Numer. math., 3, 398-412, (1961) · Zbl 0101.34101
[12]	Qiu, T.Q.; Tien, C.L., Heat transfer mechanisms during short-pulse laser heating of metals, ASME J. heat transfer, 115, 835-841, (1993)
[13]	Qiu, T.Q.; Tien, C.L., Short-pulse laser heating on metals, Internat. J. heat mass transfer, 35, 719-726, (1992)
[14]	Özisik, M.N.; Tzou, D.Y., On the wave theory in heat conduction, ASME J. heat transfer, 116, 526-536, (1994)
[15]	Tzou, D.Y., Macro to micro heat transfer, (1996), Taylor & Francis Washington DC