Accurate and stable numerical methods for solving micro heat transfer models in an n-carrier system in spherical coordinates

Energy exchange between electrons and phonons in metal provides the best example in describing non-equilibrium heating during the ultrafast transient. In times comparable to the thermalization and relaxation time of electrons and phonons, which are in the range of a few to several tens of picoseconds, heat continuously flows from hot electrons to cold phonons through mutual collisions. Consequently, electron temperature continuously decreases whereas phonon temperature continuously increases until thermal equilibrium is reached. Tien developed the well-known parabolic two-step model for describing the non-equilibrium heating in the electron-phonon system in 1992, and Tzou developed the parabolic model for the non-equilibrium heating in an N-carrier system in one-dimensional (1D) Cartesian coordinates in 2009. In the early 1990s, it was discovered that biological tissue, along with a number of other common materials, exhibits a relatively long thermal relaxation (or lag) time before equilibrium heating. Because a biological cell may contain proteins, water, and dissolved minerals, the non-equilibrium heating may also exist in the biological cell when exposed to ultrafast heating. This dissertation considers the generalized micro heat transfer models in an N-carrier system with the Neumann boundary condition in 1D and three-dimensional (3D) spherical coordinates, which can be applied to describe the non-equilibrium in biological cells. The generalized models in 1D and 3D spherical coordinates are shown to be well-posed. An improved unconditionally stable Crank-Nicholson (CN) scheme is presented for solving the generalized model in 1D spherical coordinates, where a second-order accurate finite difference scheme for the Neumann boundary condition is developed so that the overall truncation error of the 1D improved CN scheme is second-order. Two improved unconditionally stable CN schemes are then presented for solving the generalized model in 3D spherical coordinates. In particular, two second-order accurate finite difference schemes for the Neumann boundary condition are developed so that overall truncation errors of 3D improved CN schemes are second-order with respect to the spatial variable r. The stability of the 1D improved CN scheme and two 3D improved CN schemes is proved. The convergence rates of the solution of the 1D improved CN scheme are calculated by a numerical example. Results show that the convergence rates of the 1D improved CN scheme are about 2 with respect to both spatial and temporal variables respectively, while the convergence rates of the CN scheme with the convectional scheme for the Neumann boundary condition are about 1 and about 2 with respect to the spatial and temporal variables, respectively. The convergence rates of the numerical solution of two 3D improved CN schemes are calculated by two examples. Results show that the convergence rate of both 3D improved CN schemes are about 2 with respect to the spatial variable r, while the convergence rate of the 3D CN scheme is about 1 with respect to the spatial variable r.