Authors : Hameed Hasan Obaid Anous Yasaria

Volume/Issue : Volume 11 - 2026, Issue 3 - March

Google Scholar : https://tinyurl.com/3c7w3pf4

Scribd : https://tinyurl.com/35nx7p7m

DOI : https://doi.org/10.38124/ijisrt/26mar1593

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Abstract : This study addresses the challenge between global spectral accuracy and shock-capturing capability in nonlinear PDE simulations. Fourier spectral methods suffer from Gibbs phenomena near discontinuities, while finite difference methods exhibit dispersion and phase errors. A novel adaptive hybrid scheme is proposed that combines high-order finite difference operators with Fourier spectral differentiation. The method employs a spatial smoothness sensor to dynamically weight both approaches based on local solution behavior. Stability is ensured through a rigorous analysis satisfying a modified CFL condition. The scheme achieves spectral-level accuracy in smooth regions while suppressing spurious oscillations near nonsmooth areas. Numerical results demonstrate reduced L2 error and competitive computational efficiency compared to standard methods.

References :

1. Jiang, N. (2025). Research on stability and control strategies of fractional-order differential equations in nonlinear dynamic systems. Journal of Computational Methods in Sciences and Engineering, 14727978251346078.‏
2. Awrejcewicz, J., Krysko-Jr, V. A., Kalutsky, L. A., Zhigalov, M. V., & Krysko, V. A. (2021). Review of the methods of transition from partial to ordinary differential equations: From macro-to nano-structural dynamics. Archives of Computational Methods in Engineering, 28(7), 4781-4813.‏
3. Li, S., Khan, S. U., Riaz, M. B., AlQahtani, S. A., & Alamri, A. M. (2024). Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: A comprehensive stability analysis. Scientific Reports, 14(1), 6930.‏
4. Liaqat, M. I., Khan, A., Alqudah, M. A., & Abdeljawad, T. (2023). Adapted homotopy perturbation method with Shehu transform for solving conformable fractional nonlinear partial differential equations. Fractals, 31(02), 2340027.‏
5. Jalghaf, H. K., Kovács, E., Majár, J., Nagy, Á., & Askar, A. H. (2021). Explicit stable finite difference methods for diffusion-reaction type equations. Mathematics, 9(24), 3308.‏
6. Moghaddam, B. P., Babaei, A., Dabiri, A., & Galhano, A. (2024). Fractional stochastic partial differential equations: Numerical advances and practical applications—A state of the art review. Symmetry, 16(5), 563.‏
7. Liao, F., Wang, Z., Jin, Y., & Cai, J. (2025). Boundary treatments across block interfaces for high-order cell-centered finite difference method with shocks and distortion. Physics of Fluids, 37(8).‏
8. Yüksel, S. (2023). A High-order Discontinuous Galerkin Level Set Reinitialization with Finite Volume Subcell Stabilization (Master's thesis, Middle East Technical University (Turkey)).‏
9. Du, X., Dang, S., Yang, Y., & Chai, Y. (2022). The finite element method with high-order enrichment functions for elastodynamic analysis. Mathematics, 10(23), 4595.‏
10. Bürchner, T., Radtke, L., & Kopp, P. (2025). A CFL condition for the finite cell method. arXiv preprint arXiv:2502.13675.‏
11. Lucarini, S., Upadhyay, M. V., & Segurado, J. (2021). FFT based approaches in micromechanics: fundamentals, methods and applications. Modelling and Simulation in Materials Science and Engineering, 30(2), 023002.‏
12. Owolabi, K. M., Pindza, E., & Mare, E. (2025). Fourier Spectral Methods for Phase Field and Interface Dynamics: Coarsening and Pattern Formation in Energy-Based Models.‏
13. Páez-Rueda, C. I., Fajardo, A., Pérez, M., Yamhure, G., & Perilla, G. (2023). Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial. Mathematical and Computational Applications, 28(5), 93.‏
14. Brigola, R. (2025). Discrete Fourier Transforms, First Applications. In Fourier Analysis and Distributions: A First Course with Applications (pp. 85-128). Cham: Springer Nature Switzerland.‏
15. Tiwana, A. J., Zeeshan, M., Ashraf, T., Farooq, M. U., Shahzad, K., & Akhunzada, A. (2022). Dynamic link adaptation for filterband multicarrier in networks with diverse service quality and throughput requirements. Telecommunication Systems, 79(1), 109-122.‏
16. Doğan, B. K., & Karaçay, T. (2025). An investigation of service life behavior of 3D-printed hybrid polymer bearing using fused deposition modeling (FDM) method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 239(11), 4143-4151.‏