A meshless method for the numerical solution of the seventh-order Korteweg-de Vries equation
Abstract
This article describes a meshless method for the numerical solution of the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation. The meshless scheme is based on the use of the collocation method and radial basis functions. In this approach, the solution is approximated by radial basis functions, and the collocation method is used to compute the unknown coefficients. The meshless method uses the following radial basis functions: Gaussian, inverse quadratic, multiquadric, inverse multiquadric and Wu’s compactly supported radial basis function. Time discretization of the nonlinear one-dimensional non-stationary Korteweg-de Vries equation is obtained using the θ-scheme. This meshless method has an advantage over traditional numerical methods, such as the finite difference method and the finite element method, because it doesn’t require constructing an interpolation grid inside the domain of the boundary-value problem. In this meshless scheme the domain of a boundary-value problem is a set of uniformly or arbitrarily distributed nodes to which the basic functions are “tied”. The paper presents the results of the numerical solutions of two benchmark problems which were obtained using this meshless approach. The graphs of the analytical and numerical solutions for benchmark problems were obtained. Accuracy of the method is assessed in terms of the average relative error, the average absolute error, and the maximum error. Numerical experiments demonstrate high accuracy and robustness of the method for solving the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation.
Downloads
References
/References
D. J. Kortewege, and G. de Vries, “On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. 39, no. 240, pp. 422-443, 1895. doi: https://doi.org/10.1080/14786449508620739
H. Hasimoto, “Water waves” Kagaku, vol. 40, pp. 401-408, 1970. [in Japanese]
T. Kawahara, “Oscillatory Solitary Waves in Dispersive Media” Journal of the Physical Society of Japan, vol. 33, pp. 260-264, 1972. doi: https://doi.org/10.1143/JPSJ.33.260
T. Kakutani, and H. Ono, “Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma” Journal of the Physical Society of Japan, vol. 26, pp. 1305-1318, 1969. doi: https://doi.org/10.1143/JPSJ.26.1305
T. Belytschko, Y. Y. Lu and L. Gu, “Element-free Galerkin methods” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229-256, 1994. doi: https://doi.org/10.1002/nme.1620370205
T. Belytschko, Y. Rongauz and D. Organ, “Meshless methods: An overview and recently developments” Computer Methods in Applied Mechanics and Engineering, vol. 139, pp. 3-47, 1996. doi: https://doi.org/10.1016/S0045-7825(96)01078-X
M. S. Ingber, C. S. Chen, and J. A. Tanski, “A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations” International Journal for Numerical Methods in Engineering, vol. 60, no. 13, pp. 2183-2201, 2004. doi: https://doi.org/10.1002/nme.1043
I. V. Garyachevskaya, and D. O. Protektor, “Computer modeling system for the numerical solution of the one-dimensional non-stationary Burgers’ equation” Bulletin of V.N. Karazin Kharkiv National University, series «Mathematical modeling. Information technology. Automated control systems», vol. 43, pp. 11-19, 2019. doi: https://doi.org/10.26565/2304-6201-2019-43-02
D. O. Protektor, D. A. Lisin and O. Yu. Lisina, “Numerical analysis of solutions of two-dimensional heat conduction problems by meshless approach using fundamental and general solutions” Applied Questions of Mathematical Modelling, vol. 2, no. 1, pp. 98-111, 2019. doi: https://doi.org/10.32782/2618-0340-2019-3-8
D. O. Protektor, D. A. Lisin and O. Yu. Lisina, “Computer modeling system for solving three-dimensional heat conduction problems in an anisotropic environment” Radioelectronics & Informatics, vol. 84, no. 1, pp. 20-27, 2019. doi: https://doi.org/10.30837/1563-0064.1(84).2019.184712
E. J. Kansa, “Multiquadrics – A scattered data approximation scheme with applications to computational fluid-dynamics – I surface approximations and partial derivative estimates” Computers & Mathematics with Applications, vol. 19, pp. 127-145, 1990. doi: https://doi.org/10.1016/0898-1221(90)90270-T
S. G. Rubin, and R. A. Jr. Graves, “A Cubic Spline Approximation for Problems in Fluid Mechanics”, NASA Technical Reports R-436, Washington, D.C.: NASA, 1975.
W. X. Ma, “Travelling wave solutions to a seventh order generalized KdV equation” Physics Letters A, vol. 180, no. 3, pp. 221-224, 1993. doi: https://doi.org/10.1016/0375-9601(93)90699-Z
Sirendaoreji, “New exact travelling wave solutions for the Kawahara and modified Kawahara equations” Chaos, Solitons & Fractals, vol. 19, no. 1, pp. 147-150, 2004. doi: https://doi.org/10.1016/S0960-0779(03)00102-4
Kortewege D. J., de Vries G. On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 1895. Vol. 39, Issue 240. P. 422-443. doi: https://doi.org/10.1080/14786449508620739
Hasimoto H. Water waves. Kagaku. 1970. Vol. 40. P. 401-408. [in Japanese]
Kawahara T. Oscillatory Solitary Waves in Dispersive Media. Journal of the Physical Society of Japan. 1972. Vol. 33. P. 260-264. doi: https://doi.org/10.1143/JPSJ.33.260
Kakutani T., Ono H. Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma. Journal of the Physical Society of Japan. 1969. Vol. 26. P. 1305-1318. doi: https://doi.org/10.1143/JPSJ.26.1305
Belytschko T., Lu Y. Y., Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering. 1994. Vol. 37, Issue 2. P. 229-256. doi: https://doi.org/10.1002/nme.1620370205
Belytschko T., Rongauz Y., Organ D. Meshless methods: An overview and recently developments. Computer Methods in Applied Mechanics and Engineering. 1996. Vol. 139. P. 3-47. doi: https://doi.org/10.1016/S0045-7825(96)01078-X
Ingber M. S., Chen C. S., Tanski J. A. A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations. International Journal for Numerical Methods in Engineering. 2004. Vol. 60, Issue 13. P. 2183-2201. doi: https://doi.org/10.1002/nme.1043
Гарячевська І. В., Протектор Д. О. Система комп’ютерного моделювання для числового вирішення одновимірного нестаціонарного рівняння Бюргерса. Вісник Харківського національного університету імені В.Н. Каразіна, серія «Математичне моделювання. Інформаційні технології. Автоматизовані системи управління». 2019. Т. 43. С. 11-19. doi: https://doi.org/10.26565/2304-6201-2019-43-02
Протектор Д. О., Лисин Д. А., Лисина О. Ю. Численный анализ решений двумерных задач теплопроводности по бессеточной схеме с использованием фундаментальных и общих решений. Прикладні питання математичного моделювання. 2019. Т. 2, № 1. С. 98-111. doi: https://doi.org/10.32782/2618-0340-2019-3-8
Протектор Д. О., Лісін Д. О., Лісіна О. Ю. Система комп’ютерного моделювання для розв’язку тривимірних задач теплопровідності в анізотропному середовищі. Радіоелектроніка та інформатика. 2019. T. 84, № 1. С. 20-27. doi: https://doi.org/10.30837/1563-0064.1(84).2019.184712
Kansa E. J. Multiquadrics – A scattered data approximation scheme with applications to computational fluid-dynamics – I surface approximations and partial derivative estimates. Computers & Mathematics with Applications. 1990. Vol. 19. P. 127-145. doi: https://doi.org/10.1016/0898-1221(90)90270-T
Rubin S. G., Graves R. A. Jr. A Cubic Spline Approximation for Problems in Fluid Mechanics. NASA Technical Reports R-436. Washington, D.C.: NASA, 1975. 93 p.
Ma W. X. Travelling wave solutions to a seventh order generalized KdV equation. Physics Letters A. 1993. Vol. 180, Issue 3. P. 221-224. doi: https://doi.org/10.1016/0375-9601(93)90699-Z
Sirendaoreji. New exact travelling wave solutions for the Kawahara and modified Kawahara equations. Chaos, Solitons & Fractals. 2004. Vol. 19, Issue 1. P. 147-150. doi: https://doi.org/10.1016/S0960-0779(03)00102-4
