A High Order Accurate Numerical Method for Solving Two-Dimensional Dual-Phase-Lagging Equation with Temperature Jump Boundary Condition in Nanoheat Conduction

摘要

Dual-phase-lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher-order accurate and unconditionally stable compact finite difference scheme for solving one-dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two-dimensional case and develop a fourth-order accurate compact finite difference method in space coupled with the Crank-Nicolson method in time, where the Robin's boundary condition is approximated using a third-order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth-order in space and second-order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. (C) 2015 Wiley Periodicals, Inc.