Fractional parabolic two-step model and its accurate numerical scheme for nanoscale heat conduction

Abstract

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nanoscale heat conduction induced by ultrashort-pulsed laser heating. In this article, we present a nanoscale fractional parabolic two-step model and obtain an accurate numerical scheme for nanoscale heat conduction in metals. The model is obtained by introducing the Knudsen number (Kn) and the fractional-order derivative in time to the original parabolic two-step energy transport, and then coupling them with the Kn dependent and temperature-jump boundary condition. The numerical scheme is developed based on the fourth-order compact finite difference method and the L1 approximation for fractional derivatives. The well-posedness of the model, and the stability and convergence of the scheme are analyzed theoretically. We finally test the accuracy and applicability of the new model and the obtained numerical method by two examples. By changing values of the Knudsen number and fractional-order derivative as well as the parameter in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction in porous media such as porous thin metal film exposed to ultrashort-pulsed lasers.