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We present a finite-difference integration algorithm for solution of a systemof differential equations containing a diffusion equation with nonlinear terms.The approach is based on Crank-Nicolson method with predictor-correctoralgorithm and provides high stability and precision. Using a specific exampleof short-pulse laser interaction with semiconductors, we give a detaileddescription of the method and apply it for the solution of the correspondingsystem of differential equations, one of which is a nonlinear diffusionequation. The calculated dynamics of the energy density and the number densityof photoexcited free carriers upon the absorption of laser energy are presentedfor the irradiated thin silicon film. The energy conservation within 0.2% hasbeen achieved for the time step $10^4$ times larger than that in case of theexplicit scheme, for the chosen numerical setup. We also present a few examplesof successful application of the method demonstrating its benefits for thetheoretical studies of laser-matter interaction problems.

Semi-implicit finite-difference method with predictor-corrector algorithm for solution of diffusion equation with nonlinear terms