Accurate gradient preserved method for solving heat conduction equations in double layers

Abstract

Analyzing heat transfer in layered structures is important for the design and operation of devices and the optimization of thermal processing of materials. Existing numerical methods dealing with layered structures if using only three grid points across the interface usually provide only a second-order truncation error. Obtaining a numerical scheme using three grid points across the interface so that the overall numerical scheme is unconditionally stable and convergent with higher-order accuracy is mathematically challenging. In this study, we develop a higher-order accurate finite difference method using three grid points across the interface by preserving the first-order derivative, ux, in the interfacial condition and/or the boundary condition. As such, when the compact fourth-order accurate Padé scheme is used at the interior points, the overall scheme maintains to be higher-order accurate. Stability and convergence of the scheme are analyzed. Finally, four examples are given to test the obtained numerical method. Results show that the convergence order in space is close to 4.0, which coincides with the theoretical analysis.