Improving the Efficiency of Stiff ODE Solvers through Adaptive Step Size Control and Jacobian Optimization

Stiff ordinary differential equations (ODEs) present significant challenges for numerical solvers due to their multiscale nature and stringent stability requirements. Traditional methods, such as fixed-step schemes, often suffer from low computational efficiency, while fully implicit approaches may introduce numerical instability. This study proposes an improved adaptive Rosenbrock method that incorporates a PI controller-based step size adjustment mechanism and a selective Jacobian matrix update strategy to enhance both efficiency and stability. Numerical experiments on the Robertson chemical kinetics model demonstrate that, compared to conventional fixed-step solvers, the proposed method reduces the number of integration steps to just 0.4% of the original while maintaining high accuracy and significantly mitigating unphysical phenomena such as negative concentrations. These results emphasize the excellent performance of the adaptive Rosenbrock framework in solving stiff ordinary differential equation (ODE) systems. Future work may extend this approach to partial differential equations, automatic tuning of controller parameters, and stiffness prediction using machine learning techniques.