Error estimation for numerical approximations of ODEs via composition techniques. Part I: one-step methods

Abstract

In this study, we introduce a refined method for estimating errors in numerical simulations of dynamical systems through an innovative application of composition techniques. Our approach involves a dual application: a basic one-step numerical method of order p in this part, and a class of Backward Difference Formulas (BDF) schemes in Part II (Deeb et al. 2026). This dual application uses complex coefficients, resulting in outputs in the complex plane. The method’s innovation lies in demonstrating that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1 , while the imaginary parts serve as error estimates of the same order, a novel proof presented herein. The linear stability of the resulting scheme is improved over that of the basic one. The performance of the composition in computing the approximation is also compared. The results show that the proposed technique attains higher accuracy with reduced computational time relative to the basic integrators; compared with established methods of the same order it remains competitive in cost while additionally supplying a built-in error estimate, and it is most advantageous for integrators that lack a native error estimator. This dual composition technique has been rigorously applied to a variety of dynamical problems, demonstrating its efficacy in adapting the time step, particularly in situations where numerical schemes lack theoretical error estimates. Consequently, the technique has the potential to advance adaptive time-stepping strategies in numerical simulations.

Keywords

Bibliographic record

BibTeX Citation

@article{Deeb2027errorestimation,
  author = {Deeb, A. and Dutykh, D.},
  title = {Error estimation for numerical approximations of ODEs via composition techniques. Part I: one-step methods},
  journal = {Comput. Appl. Math.},
  year = {2027},
  volume = {46},
  number = {1},
  pages = {33},
  doi = {10.1007/s40314-026-03864-5},
  abstract = {In this study, we introduce a refined method for estimating errors in numerical simulations of dynamical systems through an innovative application of composition techniques. Our approach involves a dual application: a basic one-step numerical method of order p in this part, and a class of Backward Difference Formulas (BDF) schemes in Part II (Deeb et al. 2026). This dual application uses complex coefficients, resulting in outputs in the complex plane. The method’s innovation lies in demonstrating that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1 , while the imaginary parts serve as error estimates of the same order, a novel proof presented herein. The linear stability of the resulting scheme is improved over that of the basic one. The performance of the composition in computing the approximation is also compared. The results show that the proposed technique attains higher accuracy with reduced computational time relative to the basic integrators; compared with established methods of the same order it remains competitive in cost while additionally supplying a built-in error estimate, and it is most advantageous for integrators that lack a native error estimator. This dual composition technique has been rigorously applied to a variety of dynamical problems, demonstrating its efficacy in adapting the time step, particularly in situations where numerical schemes lack theoretical error estimates. Consequently, the technique has the potential to advance adaptive time-stepping strategies in numerical simulations.},
  keywords = {error estimation, numerical integration, ordinary differential equations, composition techniques, one-step methods, Runge–Kutta methods},
  publisher = {Springer Science and Business Media LLC},
  issn = {2238-3603}
}