Minimizing Rounding Errors and Improving Numerical Precision with High-Order Taylor Series Methods

This research presents an in-depth exploration of minimizing rounding errors and improving numerical precision in the numerical solution of ODEs using high-order Taylor Series Methods (TSM). Rounding errors, inherent in the IEEE 754 floating-point standard, are a universal challenge in computational mathematics as they accumulate over time, particularly in simulations of chaotic or stiff systems, leading to significant deviations from true solutions. TSM surmounts this challenge by approximating ODE solutions by high-order polynomials using Taylor series expansions, which drastically reduce truncation errors. TSM employs a set of advanced techniques, including high-precision arithmetic, automatic differentiation, compensated summation, error-free transformations, and interval arithmetic, to prevent rounding errors. These combined techniques ensure numerical stability and accuracy even in the most computationally demanding situations. The research rigorously tests the theoretical underpinnings of TSM, including full error bounds, convergence behavior, and stability analyses. Practical utility is shown through implementations like the TIDES software, which shows TSM's ability to maintain accuracy over billions of integration steps. Comparisons with traditional numerical methods, such as Runge-Kutta, multistep, and symplectic integrators, showcase TSM's better accuracy and computational efficiency for applications ranging from celestial mechanics to molecular dynamics. Further case studies, such as simulations of N-body problems and stiff biochemical systems, serve to illustrate the flexibility of TSM. Challenges, including computational intensity, memory demands, and scalability, are addressed in detail with new solutions being proposed, e.g., parallel computing frameworks, hybrid integration methods, and machine learning-based adaptivity. This expanded work positions TSM as a foundation for high-accuracy numerical analysis, offering a unifying framework for advancing computational mathematics and applications.