A new higher‐order accurate numerical method for solving heat conduction in a double‐layered film with the neumann boundary condition

Abstract

Heat conduction in multilayered films with the Neumann (or insulated) boundary condition is often encountered in engineering applications, such as laser process in a gold thin‐layer padding on a chromium thin‐layer for micromachining and patterning. Predicting the temperature distribution in a multilayered thin film is essential for precision of laser process. This article presents an accurate finite difference (FD) scheme for solving heat conduction in a double‐layered thin film with the Neumann boundary condition. In particular, the heat conduction equation is discretized using a fourth‐order accurate compact FD method in space coupled with the Crank–Nicolson method in time, where the Neumann boundary condition and the interfacial condition are approximated using a third‐order accurate compact FD method. The overall scheme is proved to be convergent and hence unconditionally stable. Furthermore, the overall scheme can be written into a tridiagonal linear system so that the Thomas algorithm can be easily used. Numerical errors and convergence rates of the solution are tested by an example. Numerical results coincide with the theoretical analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1291–1314, 2014