Two efficient numerical methods for solving Rosenau-KdV-RLW equation

Authors

DOI:

https://doi.org/10.48129/kjs.v48i1.8610

Keywords:

Cubic B-Spline functions, Galerkin method, Rosenau-KdV-RLW, Strang time-splitting

Abstract

In this study, two efficient numerical schemes based on B-spline finite element and time-splittings methods for solving Rosenau-KdV-RLW equation are going to be presented. As the first method, the equation is going to be solved by cubic B-spline Galerkin finite element method. For the second method, after splitting Rosenau-KdV-RLW equation in time, it is going to be solved by the second order Strang time-splitting technique using cubic B-spline Galerkin finite element method. The differential equation system encountered in the second method has been solved by the fourth order Runge-Kutta method. The stability analysis of the presented methods have been performed. Both methods have been applied to an example of which analytical solution is known. The numerical results obtained by the presented methods are compared with some those of the methods available in the literature by calculating the error norms L₂ and L_{∞}, convergence rates, mass and energy conservation constants. The obtained results have been found to be consistent with the compared ones.

Author Biography

Sibel Özer, İnönü University, Faculty of Arts and Sciences Department of Mathematics Malatya,TURKEY

Mathematics

Assist.Prof.Dr.

References

Rosenau, P. (1988) Dynamics of dense discrete systems. Progress of Theoretical Physics,79:1028--1042.

Park, M.A. (1993) On the Rosenau equation in multidimensional space. Nonlinear Analysis, 21:77--85.

Omrani, K., Abidi, F., Achouri, T., Khiari, N. (2008) A new conservative finite difference scheme for the Rosenau equation, Applied Mathematics and Computation.,201: 35--43.

Atouani, N., Omrani, K. (2015) A new conservative high-order accurate difference scheme for the Rosenau equation. Applicable Analysis, 94:2435--2455.

Chung, S.K, Ha, S.N. (1994) Finite element Galerkin solutions for the Rosenau equation. Applicable Analysis, 54(1-2):39-56.

Hu, B., Xu, Y., Hu, J. (2008) Crank-Nicolson finite difference scheme for the Rosenau-Burgers equation, Applied Mathematics and Computation, 204(1):311--316.

Pan, X., Zhang, L. (2012) A new finite difference scheme for the Rosenau-Burgers equation. Applied Mathematics and Computation, 218(17):8917--8924.

Janwised, J., Wongsaijai, B., Mouktonglang, T., Poochinapan, K. (2014) A Modified Three-Level Average Linear-Implicit Finite Difference Method for the Rosenau-Burgers Equation. Advances in Mathematical Physics, Article ID 734067, 11 pages. http://dx.doi.org/10.1155/2014/734067.

Piao, G.R., Lee, J.Y., Cai, G.X. (2016) Analysis and computational method based on quadratic B-spline FEM for the Rosenau-Burgers equation. Numerical Methods for Partial Differential Equations, 32(3):877-895.

Zürnacı, F., Seydaoğlu, M. (2019) On the convergence of operator splitting for the Rosenau--Burgers equation. Numerical Methods for Partial Differential Equations,35:1363--1382.https://doi.org/10.1002/num.22354.

Hu, J., Xu, Y., Hu, B. (2013) Conservative Linear Difference Scheme for Rosenau-KdV Equation, Advances in Mathematical Physics, Article ID 423718, 7 pages http://dx.doi.org/10.1155/2013/423718.

Ucar, Y., Karaagac, B., Kutluay, S. (2017) A Numerical Approach to the Rosenau-KdV equation using Galerkin Cubic Finite Element Method. International Journal of Applied Mathematics and Statistics, 56(3):83-92.

Kutluay, S., Karta, M.,Yağmurlu, N.M. (2019) Operator time-splitting techniques combined with quintic B-spline collocation method for the generalized Rosenau--KdV equation. Numerical Methods for Partial Differential Equations, 1(15). https://doi.org/10.1002/num.22409

Zuo, J.M., Zhang, Y.M., Zhang, T. D., Chang, F. (2010) A new conservative difference scheme for the general Rosenau--RLW equation. Boundary Value Problems, 13. Article ID 516260.

Pan, X., Zhang, L. (2012) On the convergence of a conservative numerical scheme for the usual Rosenau--RLW equation. Applied Mathematical Modelling, 36:3371--3378.

Atouani, N., Omrani, K. (2013) Galerkin finite element method for the Rosenau-RLW equation. Computers and Mathematics with Applications, 66:289--303.

Yagmurlu, N.M., Karaagac, B., Kutluay, S. (2017) Numerical Solutions of Rosenau-RLW Equation Using Galerkin Cubic B-Spline Finite Element Method, American Journal of Computational and Applied Mathematics, 7(1): 1-10.DOI: 10.5923/j.ajcam.20170701.01.

Foroutan, M., Ebadian, A. (2018) Chebyshev Rational Approximations for the Rosenau-KdV-RLW equation on the whole line, International Journal of Analysis and Applications,16(1):1-15.

Ghiloufi, A., Omrani, K. (2018) New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves. Numerical Methods for Partial Differential Equations, 34:451-500. https://doi.org/10.1002/num.

Ak, T., Karakoc, S.B.G., (2016) Biswas A.Numerical Scheme to Dispersive Shallow Water Waves, Journal of Computational and Theoretical Nanoscience, 13: 7084-7092. DOI: 10.1166/jctn.2016.5675

Wongsaijai, B., Poochinapan, K. (2014) A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and Rosenau-RLW equation, Applied Mathematics and Computation, 245:289-304.

Wang, X., Dai, W. (2018) A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation, Journal of Computational and Applied Mathematics, 330:295-306.

Özer, S. (2019) Numerical solution of the Rosenau--KdV--RLW equation by operator splitting techniques based on B-spline collocation method. Numerical Methods for Partial Differential Equations, 1--16. https://doi.org/10.1002/num.22387

Razborova, P., Ahmed, B., Biswas, A. (2014) Solitons, Shock Waves and Conservation Laws of Rosenau-KdV-RLW Equation with Power Law Nonlinearity. Applied Mathematics and Information Sciences, 8: 485-491.

Prenter, P. M. (1975) Splines and variational methods, John Wiley and Sons, New York.

M. K. Jain, (1983) Numerical Solution of Differential Equations, 2nd ed.,Wiley, New York, New York .

Gear, C. W. (1971) Numerical initial-value problems in ordinary differential equations, Prentice-Hall, Englewood Cliffs, NJ.

Published

2020-12-23