Design of Stable Adaptive Step Size Controller for ODE Numerical Solutions Based on Condition Number Optimization

Abstract

A numerical method to solve ordinary differential equations (ODEs) is used in computational science for many kinds of complex dynamic systems in all fields of engineering and science. Conventional adaptive step-size controllers are mainly concerned with bounding the local truncation error to maintain solution accuracy. However, for stiff systems, implicit numerical methods need to solve non-linear algebraic equations at each step and usually employ Newton-Raphson iterations. Therefore, at this time, the condition number of the iteration matrix affects both the stability and speed of convergence in practice. A new adaptive step-size controller that considers the condition number optimisation in the control loop is proposed in this paper. Continuously monitor and bound the condition number of the Jacobian-derived iteration matrix to adjust the step size dynamically in the proposed dual-loop controller, thus avoiding divergence of Newton iteration and reducing the amplification of floating-point errors. Based on the above theoretical analysis and control system construction, it has been shown that adding condition number constraints can enhance both the stability and efficiency of numerical integration for stiff ODEs. The above results provide an all-encompassing system for the development of high-accuracy and algebraically stable numerical solvers.