High-order finite difference schemes for hyperbolic conservation laws

Mudimannan, Cholaraja; Patil, Gauri; Bhan, Suraj; Bhise, Suhas; Patil, Balkrishna K.; Mounika, J.

doi:10.47974/JIM-2361

Open Access Research Article

High-order finite difference schemes for hyperbolic conservation laws

Cholaraja Mudimannandrcholaraja@clc.du.ac.inCampus Law Centre Faculty of Law University of DelhiCampus Law Centre University of Delhi New DelhiDelhi, 110007, IndiaView full profile → , ^*Gauri PatilCorresponding authorgauri.patil@dypvp.edu.inDepartment of Civil Engineering Dr. D. Y. Patil Institute of Technology Pimpri-ChinchwadPune, Maharashtra, 411018, IndiaView full profile → , Suraj Bhansuraj.bhan@niu.edu.inSchool of Engineering & Technology Noida International UniversityGreater Noida, Uttar Pradesh, 203201, IndiaView full profile → , Suhas Bhisesuhas.bhise@vit.eduDepartment of Electronics and Telecommunication Engineering Vishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile → , Balkrishna K. Patilbalkrishnapatileng@gmail.comDepartment of Computer Engineering Sandip Institute of Technology and Research Centre (SITRC) Nashik, Maharashtra, 422213, IndiaView full profile → , J. Mounikajm1995@gmail.comDepartment of Information Technology KKR and KSR Institute of Technology and SciencesDepartment of Information Technology KKR and KSR Institute of Technology and SciencesGuntur, Andhra Pradesh, 522017, IndiaView full profile →

* Corresponding author · click or hover a name for details

pp. 2197–2205Vol. 28Issue 6September 2025DOI: 10.47974/JIM-2361XML

Received:: 10 Dec 2024
Published Online:: 30 Sep 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2361
Pages:: 2197–2205

Abstract

To correctly solve hyperbolic conservation laws, which often have shocks and discontinuities, you need high-order finite difference methods. Numerical spread and instability are problems with traditional low-order methods. This is why more advanced methods like ENO and WENO were created. These methods get very accurate results in areas with flat surfaces while stopping variations near steep slopes. This makes sure that the models are strong and work well. The study talks about the basics of math, rules for stability and consistency, the structure of algorithms, and how to treat boundaries. Numerical tests show that accuracy, shock absorption, and stability gaps are all getting better.

Keywords

Hyperbolic conservation laws High-order finite difference Stability and consistency ENO/WENO schemes

Subject Classifications

65L1235L65

References

[1] Z. Zhao, Y.-T. Zhang, and J. Qiu, “A modified fifth order finite difference Hermite WENO scheme for hyperbolic conservation laws,” Journal of Scientific Computing, vol. 85, pp. 29 (2020).
[2] C. Nwaigwe and V. N. Mishra, “Accelerating fixed point algorithms for nonlinear Fredholm integral equations,” Journal of Interdisciplinary Mathematics, vol. 27, no. 6, pp. 1319–1337 (2024), doi: 10.47974/JIM-1782.
[3] B. Ren and C. Parés, “High-order WENO finite-difference methods for hyperbolic nonconservative systems of partial differential equations,” Journal of Computational Physics, vol. 535, pp. 114047 (2025).
[4] J. Li, C.-W. Shu, and J. Qiu, “Moment-based multi-resolution HWENO scheme for hyperbolic conservation laws,” Journal of Computational Physics, vol. 32, pp. 364–400 (2022).
[5] Z. Zhao and J. Qiu, “An oscillation-free Hermite WENO scheme for hyperbolic conservation laws,” Science China Mathematics, vol. 67, pp. 431–454 (2024).
[6] M. Zhang and Z. Zhao, “A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws,” Journal of Computational Physics, vol. 472, pp. 111676 (2023).
[7] I. Wibisono, Yanuar, and E. A. Kosasih, “Fifth-order Hermite targeted essentially non-oscillatory schemes for hyperbolic conservation laws,” Journal of Scientific Computing, vol. 87, pp. 69 (2021).
[8] D. Dhabliya, S. Kunche, S. Dingankar, S. S. Dari, R. Dhabliya, and V. Khetani, “Blockchain technology as a paradigm for enhancing cyber security in distributed systems,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 27, no. 2-B, pp. 729–740 (2024).
[9] P. Sahane, M. Gulhane, N. Rakesh, S. M. M. Naidu, M. Grover, and V. Mahajan, “Evaluating lightweight encryption schemes for resource-constrained devices in wireless sensor networks,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 5-A, pp. 1803–1812 (2025).
[10] M. A. A. Sheela, K. Amulya, D. Lokesh, K. Yesubabu, and P. Ajay, “Federated multimodal language recognition: A deep learning approach for real-time applications,” International Journal of Recent Advances in Engineering & Technology (IJRAET), vol. 14, no. 1, pp. 17–26 (Apr. 2025).

Views: 139Downloads: 10Citations: 0