Abstract
We present a finite-difference integration algorithm for solution of a system of differential equations containing a diffusion equation with nonlinear terms. The approach is based on Crank–Nicolson method with predictor–corrector algorithm and provides high stability and precision. Using a specific example of short-pulse laser interaction with semiconductors, we give a detailed description of the method and apply it to the solution of the corresponding system of differential equations, one of which is a nonlinear diffusion equation. The calculated dynamics of the energy density and the number density of photoexcited free carriers upon the absorption of laser energy are presented for the irradiated thin silicon film. The energy conservation within 0.2 % has been achieved for the time step 10 8 times larger than that in case of the explicit scheme, for the chosen numerical setup. The implemented Fortran source code is available in the Supplementary Materials. We also present a few examples of successful application of the method demonstrating its benefits for the theoretical studies of laser–matter interaction problems. Finally, possible extension to 2 and 3 dimensions is discussed.
Highlights
Many phenomena occurring in nature for their investigation can be described via mathematical models based on time-dependent nonlinear diffusion equations [1]
We consider an application of the nonlinear parabolic diffusion equation to describe the response of solids to an ultrashort laser pulse irradiation
The dynamics of semiconductors under the irradiation of ultrashort laser pulses can be modeled with the system of three continuum equations [17,28]: continuity equation for free carrier density and two coupled energy balance equations, one for the carriers and one for atoms:
Summary
Many phenomena occurring in nature for their investigation can be described via mathematical models based on time-dependent nonlinear diffusion equations [1]. The Crank–Nicolson semi-implicit scheme [21,22] provides unconditionally stable solution when applied to linear diffusion equations This approach is not directly applicable when nonlinear terms play an important role. We present a semi-implicit finite-difference method for the solution of a system of differential equations—one of which is a diffusion equation with nonlinear terms—and apply it to model short laser pulse interaction with silicon. The presented approach is based on Crank–Nicolson method with predictor–corrector algorithm and provides high stability and precision It has been already successfully applied for the investigation of ultrashort laser interaction with metals [23] and semiconductors [24,25,26].
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