Unveiling accurate numerical solutions of time-dependent nonlinear models via a modified hyperbolic polynomial collocation approach

Owolabi, Kolade M.; Karaagac, Berat

doi:10.22034/cmde.2025.66726.3147

Unveiling accurate numerical solutions of time-dependent nonlinear models via a modified hyperbolic polynomial collocation approach

Document Type : Research Paper

Authors

Kolade M. Owolabi ¹
Berat Karaagac ²

¹ 1. Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704, Akure, Ondo State, Nigeria. 2. Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa 0208, South Africa. 3. Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences University of the Free State, Bloemfontein 9300, South Africa.

² Department of Natural and Mathematical Sciences, Faculty of Engineering, Tarsus University, Mersin, Turkey.

10.22034/cmde.2025.66726.3147

Abstract

This paper proposes a robust numerical strategy for solving the Zeldovich combustion model by employing a hybrid method that integrates hyperbolic polynomial B-spline collocation with finite difference techniques. The Zeldovich model, which arises in combustion theory, captures complex reactive dynamics such as flame propagation, thermal explosions, and detonation waves. In the proposed scheme, time discretization is performed using a finite difference method, while the spatial discretization is handled via a Crank–Nicolson scheme for improved stability and accuracy. The inherent nonlinear terms are linearized using the Rubin–Graves technique, leading to a tractable linear system at each time step. To approximate the spatial component, fourth-order hyperbolic polynomial B-spline basis functions are employed within a collocation framework rooted in finite element methodology. The method is applied to both one-dimensional and two-dimensional versions of the Zeldovich equation. To assess its performance, the proposed approach is compared with an existing fourth-order finite difference method. Numerical experiments show that the hybrid method yields superior accuracy, particularly in capturing sharp gradients and transient dynamics. Benchmark comparisons against exact solutions confirm the method’s improved precision, with detailed error analysis provided through both $L_2$ and $L_\infty$ norms.

Keywords

Main Subjects

Numerical approximation in context of PDEs

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