A second‐order finite difference scheme for solving the dual‐phase‐lagging equation in a double‐layered nanoscale thin film

Abstract

This article considers the dual-phase-lagging (DPL) heat conduction equation in a double-layered nanoscale thin film with the temperature-jump boundary condition (i.e., Robin's boundary condition) and proposes a new thermal lagging effect interfacial condition between layers. A second-order accurate finite difference scheme for solving the heat conduction problem is then presented. In particular, at all inner grid points the scheme has the second-order temporal and spatial truncation errors, while at the boundary points and at the interfacial point the scheme has the second-order temporal truncation error and the first-order spatial truncation error. The obtained scheme is proved to be unconditionally stable and convergent, where the convergence order in L ∞ -norm is two in both space and time. A numerical example which has an exact solution is given to verify the accuracy of the scheme. The obtained scheme is finally applied to the thermal analysis for a gold layer on a chromium padding layer at nanoscale, which is irradiated by an ultrashort-pulsed laser.

Identification of the thermo-physical properties of a stratified tissue. Adiabatic hypodermic wall

“…It has been shown elsewhere [23] that the above finite-difference scheme is convergent and unconditionally stable. Moreover, the order of convergence in the L ∞ -norm is two in both space and time.…”

Determination of the thermo-physical properties of multi-layered biological tissues

“…The above FDM scheme is second-order accurate in both space and time in the L ∞norm and is unconditionally stable, [31].…”

Numerical analysis of some dual-phase-lag models

“…Corollary 11 Let the assumptions of Theorem 10 hold. Under the additional regularity (28) it follows that the approximations obtained by Problem VP 2,hk are linearly convergent; that is, there exist a positive constant C, independent of the discretization parameters h and k, such that…”

“…Even if there are numerous papers dealing with mathematical issues, to our knowledge there are few works providing the numerical simulation of these models (see, e.g., [4,5,14,20,27,28]). In this work, we revisit the models considered in [21,24,26], where the existence and uniqueness of solution was proved, and the energy decay was analyzed, deriving the necessary conditions on the time-relaxation parameters.…”