A finite difference scheme for solving a three-dimensional heat transport equation in a thin film with microscale thickness

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is di erent from the traditional heat di usion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a nite di erence scheme with two levels in time for the three-dimensional heat transport equation. It is shown by the discrete energy method that the scheme is unconditionally stable. The three-dimensional implicit scheme is then solved by using a preconditioned Richardson iteration, so that only a tridiagonal linear system is solved each iteration. Numerical results show that the solution is accurate