Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods

Abstract

Integration of Ordinary Differential Equations (ODEs) using Backward Difference formula (BDF) methods with p backward steps achieves order p accuracy if specific conditions are met. This work extends the composition technique with complex coefficients to the implicit BDF schemes, increasing the approximation order by one without additional backward points. The imaginary part of the composed flow provides an error estimate of order p + 1. Linear stability analysis reveals that the composed schemes break the Dahlquist barrier, achieving stability up to order eight. The computational performance of the composed flow outperforms BDF schemes when using the same number of backward points, allowing for higher accuracy with lower CPU time. For non-uniform meshes, the ratio of consecutive time steps, which influences stability, appears as a parameter in the roots of algebraic equations relative to the composed flow. Having a complex root with a real positive part implies a lower bound to this ratio depending on the order. For example, the bound is 0.4506 for order three and 0.6806 for order four. Numerical tests demonstrate the effectiveness of this technique in improving the accuracy and stability compared to BDF methods.

Keywords

Bibliographic record

BibTeX Citation

@article{Deeb2026errorestimation,
  author = {Deeb, A. and Dutykh, D. and Al Zohbi, M.},
  title = {Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods},
  journal = {Comput. Appl. Math.},
  year = {2026},
  volume = {45},
  number = {8},
  pages = {355},
  doi = {10.1007/s40314-026-03733-1},
  abstract = {Integration of Ordinary Differential Equations (ODEs) using Backward Difference formula (BDF) methods with p backward steps achieves order p accuracy if specific conditions are met. This work extends the composition technique with complex coefficients to the implicit BDF schemes, increasing the approximation order by one without additional backward points. The imaginary part of the composed flow provides an error estimate of order p + 1. Linear stability analysis reveals that the composed schemes break the Dahlquist barrier, achieving stability up to order eight. The computational performance of the composed flow outperforms BDF schemes when using the same number of backward points, allowing for higher accuracy with lower CPU time. For non-uniform meshes, the ratio of consecutive time steps, which influences stability, appears as a parameter in the roots of algebraic equations relative to the composed flow. Having a complex root with a real positive part implies a lower bound to this ratio depending on the order. For example, the bound is 0.4506 for order three and 0.6806 for order four. Numerical tests demonstrate the effectiveness of this technique in improving the accuracy and stability compared to BDF methods.},
  keywords = {error estimation, numerical integration, ordinary differential equations, composition techniques, backward differentiation formulas, BDF methods},
  publisher = {Springer Science and Business Media LLC},
  issn = {2238-3603}
}