An Accurate and Stable Numerical Method for Solving a Micro Heat Transfer Model in a One-Dimensional N-Carrier System in Spherical Coordinates

We consider the generalized micro heat transfer model in a 1D microsphere with N-carriers and Neumann boundary condition in spherical coordinates, which can be applied to describe nonequilibrium heating in biological cells. An accurate Crank–Nicholson type of scheme is presented for solving the generalized model, where a new second-order accurate numerical scheme for the Neumann boundary condition is developed so that the overall truncation error is second order. The scheme is proved to be unconditionally stable and convergent. The present scheme is then tested by three numerical examples. Results show that the numerical solution is much more accurate than that obtained based on the Crank–Nicholson scheme with the conventional method for the Neumann boundary condition. Furthermore, the convergence rate of the present scheme is about 1.8 with respect to the spatial variable, while the convergence rate of the Crank–Nicholson scheme with the conventional method for the Neumann boundary condition is only 1.0 with respect to the spatial variable. The scheme is ready to apply for thermal analysis in N-carrier systems.