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Peer-reviewedCC BY 4.0Mathematics

Adaptive Adams-Type Methods Based on Algebraic, Trigonometric, and Exponential Interpolation

  • Pavlina Atanasova1,2
  • Valentin Georgiev1
  • Elis Dieva1
  • Stoyan Cheresharov1
  1. 1Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria
  2. 2Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria
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Abstract

We consider an adaptive Adams-type multistep method for the numerical solution of ordinary differential equations based on algebraic, trigonometric, and exponential interpolation (the ATE method). For each interpolation class, the corresponding schemes are derived following the same construction as classical Adams methods. The stability properties of the trigonometric and exponential variants are analyzed, highlighting their differences relative to the algebraic method. The adaptive ATE algorithm selects one of the three schemes at each step according to a local interpolation indicator. This enables the method to adjust to different types of solution behavior. Numerical experiments are presented across a variety of differential equations, and the accuracy and execution times of the methods are compared.

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Cite

BibTeXRIS
BibTeX
@article{atanasova2026adaptive,
  title = {{Adaptive Adams-Type Methods Based on Algebraic, Trigonometric, and Exponential Interpolation}},
  author = {Pavlina Atanasova and Valentin Georgiev and Elis Dieva and Stoyan Cheresharov},
  journal = {Mathematics},
  year = {2026},
  month = nov,
  doi = {10.3390/math14214013},
  url = {https://www.mdpi.com/2227-7390/14/21/4013},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Adaptive Adams-Type Methods Based on Algebraic, Trigonometric, and Exponential Interpolation
AU  - Pavlina Atanasova
AU  - Valentin Georgiev
AU  - Elis Dieva
AU  - Stoyan Cheresharov
PY  - 2026
DA  - 2026/11/03
JO  - Mathematics
DO  - 10.3390/math14214013
UR  - https://www.mdpi.com/2227-7390/14/21/4013
AB  - We consider an adaptive Adams-type multistep method for the numerical solution of ordinary differential equations based on algebraic, trigonometric, and exponential interpolation (the ATE method). For each interpolation class, the corresponding schemes are derived following the same construction as classical Adams methods. The stability properties of the trigonometric and exponential variants are analyzed, highlighting their differences relative to the algebraic method. The adaptive ATE algorithm selects one of the three schemes at each step according to a local interpolation indicator. This enables the method to adjust to different types of solution behavior. Numerical experiments are presented across a variety of differential equations, and the accuracy and execution times of the methods are compared.
SN  - 2227-7390
ER  -