TY  - JOUR
AU  - Bala, Najamuddeen
AU  - Musa, Hamisu
AU  - Alhassan, Buhari
PY  - 2026
DA  - 2026/08/10
TI  - A Third-Order Variable Step Size Superclass of Block Backward Differentiation Formula for Efficient Solution of Highly Stiff Differential Systems
JO  - Journal of Nonlinear Dynamics and Applications
T2  - Journal of Nonlinear Dynamics and Applications
JF  - Journal of Nonlinear Dynamics and Applications
VL  - 2
IS  - 3
SP  - 143
EP  - 158
DO  - 10.62762/JNDA.2026.749084
UR  - https://www.icck.org/article/abs/JNDA.2026.749084
KW  - nonlinear dynamical systems
KW  - variable step size
KW  - superclass
KW  - block backward differentiation formula
KW  - convergence
AB  - This paper develops a third-order fully implicit adaptive variable step-size superclass block backward differentiation formula (VSBBDF3) for the efficient numerical simulation of nonlinear stiff dynamical systems, including oscillatory, chaotic, and reaction-kinetics systems governed by ordinary differential equations. The method extends the classical block BDF framework through a structured superclass coefficient formulation while preserving full implicitness. By computing multiple solution approximations simultaneously within each block, the scheme enhances both stability and computational efficiency. An adaptive step-size strategy controls local truncation errors, enabling dynamic response to rapidly varying stiff behavior. Rigorous theoretical analysis establishes consistency, zero-stability, convergence, and A-stability, confirming their reliability for stiff problems. Nonlinear systems from the implicit formulation are efficiently handled using Newton-type iteration. Extensive numerical experiments on stiff linear, oscillatory, and nonlinear problems demonstrate that the proposed method consistently achieves higher accuracy with competitive computational cost compared to existing methods, including NBDF, VSBBDF, and MATLAB ODE solvers. The results further show that combining block formulation, superclass structure, and adaptive step-size control provides a more effective accuracy-efficiency balance. Overall, the VSBBDF3 method offers a robust, accurate, and efficient framework for the numerical simulation of nonlinear stiff dynamical systems, well suited for scientific and engineering applications in dynamics, control, and bifurcation analysis.
SN  - 3069-6313
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Bala2026A,
  author = {Najamuddeen Bala and Hamisu Musa and Buhari Alhassan},
  title = {A Third-Order Variable Step Size Superclass of Block Backward Differentiation Formula for Efficient Solution of Highly Stiff Differential Systems},
  journal = {Journal of Nonlinear Dynamics and Applications},
  year = {2026},
  volume = {2},
  number = {3},
  pages = {143-158},
  doi = {10.62762/JNDA.2026.749084},
  url = {https://www.icck.org/article/abs/JNDA.2026.749084},
  abstract = {This paper develops a third-order fully implicit adaptive variable step-size superclass block backward differentiation formula (VSBBDF3) for the efficient numerical simulation of nonlinear stiff dynamical systems, including oscillatory, chaotic, and reaction-kinetics systems governed by ordinary differential equations. The method extends the classical block BDF framework through a structured superclass coefficient formulation while preserving full implicitness. By computing multiple solution approximations simultaneously within each block, the scheme enhances both stability and computational efficiency. An adaptive step-size strategy controls local truncation errors, enabling dynamic response to rapidly varying stiff behavior. Rigorous theoretical analysis establishes consistency, zero-stability, convergence, and A-stability, confirming their reliability for stiff problems. Nonlinear systems from the implicit formulation are efficiently handled using Newton-type iteration. Extensive numerical experiments on stiff linear, oscillatory, and nonlinear problems demonstrate that the proposed method consistently achieves higher accuracy with competitive computational cost compared to existing methods, including NBDF, VSBBDF, and MATLAB ODE solvers. The results further show that combining block formulation, superclass structure, and adaptive step-size control provides a more effective accuracy-efficiency balance. Overall, the VSBBDF3 method offers a robust, accurate, and efficient framework for the numerical simulation of nonlinear stiff dynamical systems, well suited for scientific and engineering applications in dynamics, control, and bifurcation analysis.},
  keywords = {nonlinear dynamical systems, variable step size, superclass, block backward differentiation formula, convergence},
  issn = {3069-6313},
  publisher = {Institute of Central Computation and Knowledge}
}