Cubic B-Spline Quasi-Interpolation Method For Regularized Long Wave Equation
Yıl 2018,
Cilt: 3 Sayı: 3, 173 - 187, 29.12.2018
Dursun Irk
,
Mehmet Ali Mersin
Öz
In this study, we present a numerical method to solve the Regularized Long Wave (RLW) equation,
based on cubic B-spline quasi-interpolation for the space integration and Crank-Nicolson method
for the time integration. The method is tested on the problems of propagation of a solitary wave
and interaction of two solitary waves. The three conservation quantities of the motion are calculated
to determine the conservation properties of the proposed algorithm.
Kaynakça
- [1] Peregrine, D.H., “Calculations of the development of an undular bore”, J. Fluid. Mech. 25 (2) (1966): 321–330.
- [2] Benjamin, T.B., Bona, J.L., and Mahony, J.J., “Model equations for long waves in non-linear dispersive systems”, Philos. Trans. R. Soc., London A 272 (1972): 47–78.
- [3] Eilbeck, J.C., and McGuire, G.R. “Numerical study of RLW equation” I: numerical methods”, J. Comput. Phys. 19 (1975) 43–57.
- [4] Eilbeck, J.C. and McGuire, G. R., “Numerical study of the regularized long-wave equation II: interaction of solitary waves”, Journal of Computational Physics 23, (1977): 63-73.
- [5] Padam, C.J. and Iskandar, L. “Numerical solutions of the regularized long wave equation”, Comp. Methods Appl. Mech. Eng. 20 (1979): 195–201.
- [6] Irk, D., Dag, I. and Dogan, A., “Numerical integration of the RLW equation using cubic splines”, Anziam Journal 47, (2005):131-142.
- [7] Raslan, K.R., “A computational method for the regularized long wave (RLW) equation”, Applied Mathematics and Computation 167(2), (2005b):1101–1118.
- [8] Soliman, A. A. and Hussien, M. H., “Collocation solution for RLW equation with septic spline”, Applied Mathematics and Computation 161, (2005):623–636.
- [9] Saka, B., Dag, I. and Irk, D., “Quintic B-spline collocation method for numerical solution of the RLW equation”, Anziam Journal 49, (2008b):389-410.
- [10] Saka, B., Sahin, A. and Dag, I., “B-spline collocation algorithms for numerical solution of the RLW equation”, Wiley Subscription Services, Inc., A Wiley Company 27, 3, (2011):581-607.
- [11] Irk, D., “Solitary wave solutions for the regularized long-wave equation”, Physics of Wave Phenomena 20, 3, (2012):174-183.
- [12] Saka, B., and Dag, I. “A collocation method for the numerical solution of the RLW equation using cubic B-spline basis”, Arabian Journal for Science and Engineering. 30. 39-50, (2005).
- [13] Esen, A. and Kutluay, S., “Application of a lumped Galerkin method to the regularized long wave equation”, Applied Mathematics and Computation 174, (2006):833–845.
- [14] Alexander, M.E. and Morris, J.LL. “Galerkin methods applied to some model equations for nonlinear dispersive waves”, J. Comput. Phys. 30 (1979): 428–451.
- [15] Gardner, L.R.T. and Gardner, G.A. “Solitary wave of the regularised long wave equation”, J. Comput. Phys. 91 (1990): 441–459.
- [16] Gardner, L.R.T., Gardner, G.A. and Dag, I. “A B-spline finite element method for the regularized long wave equation”, Commun. Numer., Methods Eng. 11 (1995): 59–68.
- [17] Gardner, L.R.T. and I. Dag, I. “The boundary-forced regularised long-wave equation”, Il Nuo. Cimen. 110B (12), (1995): 1487–1495.
- [18] Gardner, L.R.T., Gardner, G.A. and Dogan, A. “A least squares finite element scheme for the RLW equation”, Commun. Numer., Methods Eng. 12 (1996): 795–804.
- [19] Dag, I. and Ozer, M.N. “Approximation of RLW equation by least square cubic B-spline finite element method”, Appl. Math. Model. 25 (2001): 221–231.
- [20] Dogan, A., “Numerical solution of regularized long wave equation using Petrov-Galerkin method”, Communications in Numerical Methods in Engineering 17, (2001):485–494.
- [21] Dogan, A., “Numerical solution of RLW equation using linear finite elements within Galerkin’s method”, Applied Mathematical Modelling 26(7), (2002):771-783.
- [22] Saka, B. and Dag, I., “A numerical solution of the RLW equation by Galerkin method using quartic B-splines”, Communications in Numerical Methods in Engineering 24, (2008):1339-1361.
- [23] Dag, I., Korkmaz A. and Saka, B., “Cosine expansion-based differential quadrature algorithm for numerical solution of the RLW equation” 26, 3, (2010):544-560.
- [24] Korkmaz, A. and Dag, I. “Numerical Simulations of Boundary-Forced RLW Equation with Cubic B-Spline-based Differential Quadrature Methods”, Arabian Journal for Science and Engineering, Volume 38, Issue 5, pp (2013):1151–1160.
- [25] Mersin M.A. “B-spline Quasi-Interpolation Method for Numerical Solutions of some Partial Differential Equations”, M.S., Eskişehir Osmangazi University, (2014), 80p.
- [26] Farin, G., “Curves and Surfaces for CAGD”, fifth ed., Morgan Kaufman, San Francisco, (2001).
- [27] Sablonnière, P., “Quasi-interpolants splines sobre particiones uniforms, in: First Meeting in Approximation Theory of the University of Jaén, Ubeda” Prépublication IRMAR 00-38, Rennes, (June 29–July 2, 2000).
- [28] Sablonnière, P., “Univariate spline quasi-interpolants and applications to numerical analysis”, Rend. Sem. Mat. Univ. Pol. Torino 63, (2005):211–222.
- [29] Zhu, C. G. and Kang, W. S., “Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation”, Applied Mathematics and Computation 216, (2010):2679-2686.
- [30] Rubin S. G. and Graves R. A., “A Cubic Spline Approximation for Problems in Fluid Mechanics,” Nasa TR R-436, Washington, DC, (1975).
- [31] Olver, P.J. “Euler operators and conservation laws of the BBM equation”, Math. Proc. Camb. Philos. Soc. 85 (1979): 143–159.
Yıl 2018,
Cilt: 3 Sayı: 3, 173 - 187, 29.12.2018
Dursun Irk
,
Mehmet Ali Mersin
Kaynakça
- [1] Peregrine, D.H., “Calculations of the development of an undular bore”, J. Fluid. Mech. 25 (2) (1966): 321–330.
- [2] Benjamin, T.B., Bona, J.L., and Mahony, J.J., “Model equations for long waves in non-linear dispersive systems”, Philos. Trans. R. Soc., London A 272 (1972): 47–78.
- [3] Eilbeck, J.C., and McGuire, G.R. “Numerical study of RLW equation” I: numerical methods”, J. Comput. Phys. 19 (1975) 43–57.
- [4] Eilbeck, J.C. and McGuire, G. R., “Numerical study of the regularized long-wave equation II: interaction of solitary waves”, Journal of Computational Physics 23, (1977): 63-73.
- [5] Padam, C.J. and Iskandar, L. “Numerical solutions of the regularized long wave equation”, Comp. Methods Appl. Mech. Eng. 20 (1979): 195–201.
- [6] Irk, D., Dag, I. and Dogan, A., “Numerical integration of the RLW equation using cubic splines”, Anziam Journal 47, (2005):131-142.
- [7] Raslan, K.R., “A computational method for the regularized long wave (RLW) equation”, Applied Mathematics and Computation 167(2), (2005b):1101–1118.
- [8] Soliman, A. A. and Hussien, M. H., “Collocation solution for RLW equation with septic spline”, Applied Mathematics and Computation 161, (2005):623–636.
- [9] Saka, B., Dag, I. and Irk, D., “Quintic B-spline collocation method for numerical solution of the RLW equation”, Anziam Journal 49, (2008b):389-410.
- [10] Saka, B., Sahin, A. and Dag, I., “B-spline collocation algorithms for numerical solution of the RLW equation”, Wiley Subscription Services, Inc., A Wiley Company 27, 3, (2011):581-607.
- [11] Irk, D., “Solitary wave solutions for the regularized long-wave equation”, Physics of Wave Phenomena 20, 3, (2012):174-183.
- [12] Saka, B., and Dag, I. “A collocation method for the numerical solution of the RLW equation using cubic B-spline basis”, Arabian Journal for Science and Engineering. 30. 39-50, (2005).
- [13] Esen, A. and Kutluay, S., “Application of a lumped Galerkin method to the regularized long wave equation”, Applied Mathematics and Computation 174, (2006):833–845.
- [14] Alexander, M.E. and Morris, J.LL. “Galerkin methods applied to some model equations for nonlinear dispersive waves”, J. Comput. Phys. 30 (1979): 428–451.
- [15] Gardner, L.R.T. and Gardner, G.A. “Solitary wave of the regularised long wave equation”, J. Comput. Phys. 91 (1990): 441–459.
- [16] Gardner, L.R.T., Gardner, G.A. and Dag, I. “A B-spline finite element method for the regularized long wave equation”, Commun. Numer., Methods Eng. 11 (1995): 59–68.
- [17] Gardner, L.R.T. and I. Dag, I. “The boundary-forced regularised long-wave equation”, Il Nuo. Cimen. 110B (12), (1995): 1487–1495.
- [18] Gardner, L.R.T., Gardner, G.A. and Dogan, A. “A least squares finite element scheme for the RLW equation”, Commun. Numer., Methods Eng. 12 (1996): 795–804.
- [19] Dag, I. and Ozer, M.N. “Approximation of RLW equation by least square cubic B-spline finite element method”, Appl. Math. Model. 25 (2001): 221–231.
- [20] Dogan, A., “Numerical solution of regularized long wave equation using Petrov-Galerkin method”, Communications in Numerical Methods in Engineering 17, (2001):485–494.
- [21] Dogan, A., “Numerical solution of RLW equation using linear finite elements within Galerkin’s method”, Applied Mathematical Modelling 26(7), (2002):771-783.
- [22] Saka, B. and Dag, I., “A numerical solution of the RLW equation by Galerkin method using quartic B-splines”, Communications in Numerical Methods in Engineering 24, (2008):1339-1361.
- [23] Dag, I., Korkmaz A. and Saka, B., “Cosine expansion-based differential quadrature algorithm for numerical solution of the RLW equation” 26, 3, (2010):544-560.
- [24] Korkmaz, A. and Dag, I. “Numerical Simulations of Boundary-Forced RLW Equation with Cubic B-Spline-based Differential Quadrature Methods”, Arabian Journal for Science and Engineering, Volume 38, Issue 5, pp (2013):1151–1160.
- [25] Mersin M.A. “B-spline Quasi-Interpolation Method for Numerical Solutions of some Partial Differential Equations”, M.S., Eskişehir Osmangazi University, (2014), 80p.
- [26] Farin, G., “Curves and Surfaces for CAGD”, fifth ed., Morgan Kaufman, San Francisco, (2001).
- [27] Sablonnière, P., “Quasi-interpolants splines sobre particiones uniforms, in: First Meeting in Approximation Theory of the University of Jaén, Ubeda” Prépublication IRMAR 00-38, Rennes, (June 29–July 2, 2000).
- [28] Sablonnière, P., “Univariate spline quasi-interpolants and applications to numerical analysis”, Rend. Sem. Mat. Univ. Pol. Torino 63, (2005):211–222.
- [29] Zhu, C. G. and Kang, W. S., “Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation”, Applied Mathematics and Computation 216, (2010):2679-2686.
- [30] Rubin S. G. and Graves R. A., “A Cubic Spline Approximation for Problems in Fluid Mechanics,” Nasa TR R-436, Washington, DC, (1975).
- [31] Olver, P.J. “Euler operators and conservation laws of the BBM equation”, Math. Proc. Camb. Philos. Soc. 85 (1979): 143–159.