Using automatic differentiation to solve concentration dependent diffusion problems

摘要

We present a Taylor series method for the solution of a class of nonlinear diffusion problems involving a concentration dependent diffusion coefficient. The computational initial-boundary value problem is to determine the concentration of the diffusing substance in a semi-infinite domain at any time, starting with a given initial concentration. The method of solution begins by first mapping the semi-infinite physical domain to a finite computational domain. Then the solution at each spatial grid point is advanced in time using a Taylor series expansion. The method employs a technique known as automatic differentiation, which is neither numerical nor symbolic. This technique is the evaluation of the coefficients in a Taylor series expansion using recursive formulas derived from the differential equation describing the initial value problem. The results obtained using the method of this paper is are in excellent agreement with approximate similarity solutions obtained previously.