Numerical Solving Strategy for High-Energy Laser Thermal Management Based on the Nonlinear Heat Conduction Equation

High-energy laser systems are fundamental to modern advanced manufacturing and defense engineering, where effective thermal management is the key bottleneck limiting beam quality and operational longevity. This paper presents a robust numerical framework for simulating transient temperature fields in laser gain media, governed by a nonlinear parabolic partial differential equation (PDE). We first establish a comprehensive thermophysical model incorporating conductive heat transfer, convective boundary cooling, and crucially, the temperature-dependent thermal conductivity of the solid-state crystal (e.g., YAG or Sapphire). This results in a nonlinear heat conduction equation. For its numerical solution, an implicit Finite Difference Method (FDM) scheme is implemented, ensuring unconditional stability for the stiff problem arising from intense internal heat generation. A predictor-corrector iteration is designed to handle the nonlinearity introduced by the variable conductivity. Applied to a simplified cylindrical laser rod geometry, the model successfully simulates the evolution of temperature and thermal stress under pulsed operation. The numerical solution reveals a significant thermal lensing effect, with a maximum temperature rise of 152.3 K and a corresponding focal power of 0.85 m⁻¹. Comparative analysis demonstrates that neglecting the nonlinearity (assuming constant conductivity) leads to a 12.7% underestimation of the peak temperature and a mischaracterization of the stress profile. This work provides a reliable and adaptable computational tool for the design and optimization of thermal management systems in high-power laser engineering, highlighting the critical role of applied mathematics in tackling complex multiphysics challenges.