Öz
Coupled Korteweg de Vries KdV equations in Hamiltonian form are integrated by the energy preserving average vector field AVF method. Numerical results confirm long term preservation of the energy and the quadratic invariants. Produced generalized solitary waves are similar to those in the literature for larger mesh sizes and time steps. Numerical and continuous dispersion relations of the linearized equations are compared to analyze the behavior of the traveling waves and the interaction of the solitons.
Anahtar Kelimeler
Energy preservation bi-Hamiltonian systems Poisson structure KdV equa
Kaynakça
- Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255–269.
- Ascher, U. M. and McLachlan, R. I., (2004), On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25, 83–104.
- Aydın, A. and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modiŞed Korteweg-de Vries equation, J. Math. Phys., 51, 083511.
- Bona, J. L, Dougalis, V, A, and Mitsotakis, D. E., (2007), Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves, Math. Comput. Simulation, 74, 214-228.
- Bona, J. L., Dougalis V. A., and Mitsotakis, D. E,, (2008), Numerical solution of Boussinesq systems of KdV-KdV type. II. Evolution of radiating solitary waves, Nonlinearity, 21, 2825-2848.
- Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51, 91–100.
- Dahlby, M., (2011), A General Framework for Deriving Integral Preserving Numerical Methods for PDEs, SIAM Journal on ScientiŞc Computing, 33, 2318–2340.
- Guha, P., (2005), Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems, J. Math. Anal. Appl., 310, 45-56.
- Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73–84.
- Hairer, E., Lubich, C. and Wanner, G., (2006), Geometric Numerical Integration-Structure- Preserving Algorithms for Ordinary Differential Equations, Springer.
- Olver, P., (1995), Applications of Lie Groups to Differential Equations, second edition, Springer.
Öz
Kaynakça
- Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255–269.
- Ascher, U. M. and McLachlan, R. I., (2004), On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25, 83–104.
- Aydın, A. and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modiŞed Korteweg-de Vries equation, J. Math. Phys., 51, 083511.
- Bona, J. L, Dougalis, V, A, and Mitsotakis, D. E., (2007), Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves, Math. Comput. Simulation, 74, 214-228.
- Bona, J. L., Dougalis V. A., and Mitsotakis, D. E,, (2008), Numerical solution of Boussinesq systems of KdV-KdV type. II. Evolution of radiating solitary waves, Nonlinearity, 21, 2825-2848.
- Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51, 91–100.
- Dahlby, M., (2011), A General Framework for Deriving Integral Preserving Numerical Methods for PDEs, SIAM Journal on ScientiŞc Computing, 33, 2318–2340.
- Guha, P., (2005), Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems, J. Math. Anal. Appl., 310, 45-56.
- Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73–84.
- Hairer, E., Lubich, C. and Wanner, G., (2006), Geometric Numerical Integration-Structure- Preserving Algorithms for Ordinary Differential Equations, Springer.
- Olver, P., (1995), Applications of Lie Groups to Differential Equations, second edition, Springer.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Bölüm | Research Article |
| Yazarlar | |
| Yayımlanma Tarihi | 1 Aralık 2012 |
| Yayımlandığı Sayı | Yıl 2012 Cilt: 2 Sayı: 2 |