Numerical Solution of GRLW Equation by Cubic Hermit B-Spline Collocation Method

Abstract

This study deals with the numerical solution of the Generalized Regular Long Wave (GRLW) equation, which holds an important place in fluid dynamics. To this end, first of all, the GRLW equation is discretized by using the Crank-Nicolson method, then linearized using the Rubin-Graves linearization technique for nonlinear terms, and a linear numerical scheme is obtained by applying the cubic Hermite B-spline collocation finite element method. This proposed numerical scheme is applied to a solitary wave problem and the interaction of two and three solitary waves. Conservation constants and the absolute change quantities of the conservation constants are calculated for all problems, the error norms for a single solitary wave problem which has an exact analytical solution. The calculated error norms and the conservation constants are presented in tables and compared with those of the existing studies for the same parameters to test the accuracy of the method. Furthermore, the graphs of the obtained numerical results are drawn to visually present the physical behavior of the problems. The tables and graphs show that the proposed numerical scheme sufficiently preserves the conservation constants of the problems and is in good agreement with the existing studies.

Keywords

• GRLW equation
• collocation method
• cubic Hermite B-spline

Supporting Institution

The authors declare that they received no financial support for the research, authorship, and/or publication of this article.

Ethical Statement

The work does not require ethics committee approval and any private permission.

GRLW Denkleminin Kübik Hermite B-splayn Kollakasyon Yöntemiyle Sayısal Çözümü

Abstract

Bu çalışma, akışkanlar dinamiğinde önemli bir yere sahip olan Genelleştirilmiş Düzenli Uzun Dalga (GRLW) denkleminin sayısal çözümünü içermektedir. Bu amaçla, GRLW denklemi Crank-Nicolson yöntemi ile ayrıklaştırıldıktan sonra, doğrusal olmayan terim Rubin-Graves doğrusallaştırma tekniğiyle lineerleştirildi ve kübik Hermite B-splayn kollakasyon sonlu elemanlar yöntemi uygulanarak bir lineer sayısal şema elde edildi. Bu önerilen sayısal şema bir soliter dalga problemi ile iki ve üç soliter dalganın girişimi problemlerine uygulanarak korunum sabitleri ve korunum sabitlerinin mutlak değişim miktarları ve ayrıca analitik çözümü mevcut olan bir soliter dalga problemi için hata normları hesaplandı. Hesaplanan hata normları ve korumum sabitleri tablolar halinde sunuldu ve aynı parametreler için mevcut olan çalışmalarla karşılaştırılarak yöntemin doğruluğu test edildi. Ayrıca, elde edilen sayısal sonuçların grafikleri çizilerek problemlerin fiziksel davranışları görsel olarak sunuldu. Tablolar ve grafikler, önerilen sayısal şemanın problemlerin korunum sabitlerini yeterince iyi bir şekilde koruduğunu ve mevcut olan çalışmalarla oldukça uyumlu olduğunu göstermektedir.

Keywords

• GRLW denklemi
• kollakasyon yöntemi
• kübik Hermite B-splayn

Supporting Institution

Yazarlar ara¸stırma, yazarlık ya da çalı¸smanın yayınlanması için herhangi bir finansman destek almadıklarını beyan eder.

Ethical Statement

Çalışma, etik kurul izni veya herhangi bir özel izin gerektirmemektedir.

References

1. Peregrine, D. H. (1966). Calculations of the development of an undular bore. Journal of Fluid Mechanics, 25, 321-330. https://doi.org/10.1017/S0022112066001678
2. Peregrine, D. H. (1967). Long waves on a beach. Journal of Fluid Mechanics, 27(4), 815-827. https://doi.org/10.1017/S0022112067002605
3. Zhang, L. (2005). A finite difference scheme for generalized regularized long-wave equation. Applied Mathematics and Computation, 168(2), 962-972. https://doi.org/10.1016/j.amc.2004.09.027
4. Ramos, J. I. (2007). Solitary wave interactions of the GRLW equation. Chaos, Solitons and Fractals, 33(2), 479-491. https://doi.org/10.1016/j.chaos.2006.01.016
5. Mokhtari, R., & Mohammadi, M. (2010). Numerical solution of GRLW equation using Sinc-collocation method. Computer Physics Communications, 181(7), 1266-1274. https://doi.org/10.1016/j.cpc.2010.03.015
6. Wang, J. F., Bai, F. N., & Cheng, Y. M. (2011). A meshless method for the nonlinear generalized regularized long wave equation. Chinese Physics B, 20(3), Article 030206. https://iopscience.iop.org/article/10.1088/1674-1056/20/3/030206
7. Roshan, T. (2012). A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation. Computers and Mathematics with Applications, 63(5), 943-956. https://doi.org/10.1016/j.camwa.2011.11.059
8. El-Danaf, T. S., Raslan, K. R., & Ali, K. K. (2014). New numerical treatment for the generalized regularized long wave equation based on finite difference scheme. International Journal of Soft Computing and Engineering, 4(4), 16-24. https://www.ijsce.org/portfolio-item/D2328094414/

9. Berikelashvili, G., & Mirianashvili, M. (2014). On the convergence of difference schemes for generalized Benjamin-Bona-Mahony equation. Numerical Methods for Partial Differential Equations, 30(1), 301-320. https://doi.org/10.1002/num.21810
10. Akbari, R., & Mokhtari, R. (2014). A new compact finite difference method for solving the generalized long wave equation. Numerical Functional Analysis and Optimization, 35(2), 133-152. https://doi.org/10.1080/01630563.2013.830128
11. Guo, P. F., Zhang, L. W., & Liew, K. M. (2014). Numerical analysis of generalized regularized long wave equation using the element-free KP-Ritz method. Applied Mathematics and Computation, 240, 91-101. https://doi.org/10.1016/j.amc.2014.04.023
12. Hammad, D. A., & El-Azab, M. S. (2015). A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation. Applied Mathematics and Computation, 253, 248-261. https://doi.org/10.1016/j.amc.2014.12.070
13. García-López, C. M., & Ramos, J. I. (2015). Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations. Applied Mathematical Modelling, 39(21), 6645-6668. https://doi.org/10.1016/j.apm.2015.02.016
14. Pan, X., Che, H., & Wang, Y. (2015). A high-accuracy compact conservative scheme for generalized regularized long-wave equation. Boundary Value Problems, 2015, Article 141. https://doi.org/10.1186/s13661-015-0404-7
15. Mohammadi, R. (2015). Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation. Chinese Physics B, 24(5), Article 050206. https://doi.org/10.1088/1674-1056/24/5/050206
16. Li, S., Wang, J., & Luo, Y. (2015). A Fourth-Order conservative compact finite difference scheme for the generalized RLW equation. Mathematical Problems in Engineering, 2015(1), Article 960602. https://doi.org/10.1155/2015/960602
17. Hammad, D. A., & El-Azab, M. (2016). Chebyshev-Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation. Applied Mathematics and Computation, 285, 228-240. https://doi.org/10.1016/j.amc.2016.03.033
18. Karakoç, S. B. G., & Zeybek, H. (2016). Solitary-wave solutions of the GRLW equation using septic B-spline collocation method. Applied Mathematics and Computation, 289, 159-171. https://doi.org/10.1016/j.amc.2016.05.021
19. Zeybek, H., & Karakoç, S. B. G. (2016). A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline. SpringerPlus, 5(1), Article 199. https://doi.org/10.1186/s40064-016-1773-9
20. Ramos, J. I. (2016). On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions. International Journal of Numerical Methods for Heat & Fluid Flow, 26(3), 698-721. https://doi.org/10.1108/HFF-07-2015-0288
21. El-Danaf, T. S., Raslan, K. R., & Ali, K. K. (2017). Non-polynomial spline method for solving the generalized regularized long wave equation. Communication in Mathematical Modeling and Applications, 2(2), 1-17. https://dergipark.org.tr/tr/pub/cmma/issue/35978/403634
22. Ramos, J. I., & García-López, C. M. (2017). Time-linearized, compact methods for the inviscid GRLW equation subject to initial Gaussian conditions. Applied Mathematical Modelling, 48, 353-383. https://doi.org/10.1016/j.apm.2017.04.014
23. Li, Q., & Mei, L. (2018). Local momentum-preserving algorithms for the GRLW equation. Applied Mathematics and Computation, 330, 77-92. https://doi.org/10.1016/j.amc.2018.02.033
24. Bhowmik, S. K., & Karakoç, S. B. (2019). Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method. Numerical Methods for Partial Differential Equations, 35(6), 2236-2257. https://doi.org/10.1002/num.22410
25. Zeybek, H., & Karakoç, S. (2019). A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation. Scientia Iranica, 26(6), 3356-3368. https://doi.org/10.24200/sci.2018.20781
26. Alharbi, A. R. (2020). Numerical investigation for the GRLW equation using parabolic Monge Ampere equation. International Journal of Mathematics and Computer Science, 15(2), 443-462. https://future-in-tech.net/15.2/R-Alharbi.pdf
27. Rouatbi, A., Labidi, M., & Omrani, K. (2020). Conservative difference scheme of solitary wave solutions of the generalized regularized long-wave equation. Indian Journal of Pure and Applied Mathematics, 51(4), 1317-1342. https://doi.org/10.1007/s13226-020-0468-7
28. Oruç, Ö. (2020). Numerical investigation of nonlinear generalized regularized long wave equation via delta-shaped basis functions. An International Journal of Optimization and Control: Theories and Applications, 10(2), 244-258. https://doi.org/10.11121/ijocta.01.2020.00881
29. Hammad, D. A. (2021). Application of Bernstein collocation method for solving the generalized regularized long wave equations. Ain Shams Engineering Journal, 12(4), 4081-4089. https://doi.org/10.1016/j.asej.2021.04.005
30. Almatrafi, M. B., Alharbi, A., Lotfy, K., & El-Bary, A. A. (2021). Exact and numerical solutions for the GBBM equation using an adaptive moving mesh method. Alexandria Engineering Journal, 60(5), 4441-4450. https://doi.org/10.1016/j.aej.2021.03.023
31. Karakoç, S. B. G., Mei, L., & Ali, K. K. (2022). Two efficient methods for solving the generalized regularized long wave equation. Applicable Analysis, 101(13), 4721-4742. https://doi.org/10.1080/00036811.2020.1869942
32. Al-Humedi, H. O. (2022). Combining B-spline least-square schemes with different weight functions to solve the generalized regularized long wave equation. International Journal of Nonlinear Analysis and Applications, 13(1), 159-177. https://doi.org/10.22075/ijnaa.2022.5468
33. Behnood, M., & Shokri, A. (2022). A Legendre spectral element method for the family of regularized long wave equations. Mathematics and Computers in Simulation, 201, 239-253. https://doi.org/10.1016/j.matcom.2022.05.019
34. Shallu, & Kukreja, V. K. (2023). Solution of the generalized regularized long-wave equation with optimal spline collocation technique and implicit Crank-Nicolson as well as explicit SSP-RK43 scheme. International Journal of Computer Mathematics, 100(1), 1-19. https://doi.org/10.1080/00207160.2022.2078662
35. Kumari, A., & Kukreja, V. K. (2023). Study of generalized regularized long wave equation via septic Hermite collocation method with Crank-Nicolson and SSP-RK43 schemes to capture the various solitons. Wave Motion, 122, Article 103188. https://doi.org/10.1016/j.wavemoti.2023.103188
36. Karta, M. (2005). More efficient solutions for numerical analysis of the nonlinear generalized regularized long wave (GRLW) using the operator splitting method. Fundamental Journal of Mathematics and Applications, 8(2), 72-87. https://doi.org/10.33401/fujma.1461430
37. Oruç, Ö. (2025). An extrapolated second-order BDF combined with local meshfree radial point interpolation method for numerical simulation of multi-dimensional regularized long wave equation emerging in fluids. Journal of Scientific Computing 104(44), 1-32. https://doi.org/10.1007/s10915-025-02948-4
38. Oruç, Ö. (2025). A barycentric Floater-Hormann interpolation with time splitting for numerical simulation of multi-dimensional regularized long wave equation. Computational and Applied Mathematics 44(225), 1-28. https://doi.org/10.1007/s40314-025-03183-1
39. Prenter, P. M. (1989). Splines and variational methods. John Wiley & Sons Inc.
40. de Boor, C. (1978). A practical guide to splines. Springer Verlag.
41. Kumari, A., & Kukreja, V. K. (2023). Survey of Hermite interpolating polynomials for the solution of differential equations. Mathematics, 11(14), 3157. https://doi.org/10.3390/math11143157
42. Rubin, S. G., & Graves R. A. (1975). A cubic spline approximation for problems in fluid mechanics. (Report No. NASA TR R-436). National Aeronautics and Space Administration.
43. Rashidinia, J., & Rasoulizadeh, M. N. (2019). Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation. Wave Motion, 90, 152-167. https://doi.org/10.1016/j.wavemoti.2019.05.006
44. Biswas, A. (2010). Solitary waves for power-law regularized long-wave equation and $R(m,n)$ equation. Nonlinear Dynamics, 59, 423-426. https://doi.org/10.1007/s11071-009-9548-2