Geometric numerical schemes for the KdV equation

Denys Dutykh (1, 2, 3) , Marx Chhay (4) , Francesco Fedele (5, 6)

1 LAMA - Laboratoire de Mathématiques
2 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 LOCIE - LabOratoire proCédés énergIe bâtimEnt
5 School of Civil and Environmental Engineering [Atlanta]
6 ECE GeorgiaTech - School of Electrical and Computer Engineering - Georgia Insitute of Technology

Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.

Mots clés: KdV equation; symplectic scheme; multi-symplectic scheme; solitonic gas; wave turbulence

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Physique Numérique [physics.comp-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]; Dynamique des Fluides [physics.flu-dyn]; Analyse numérique [math.NA]

Denys Dutykh, Marx Chhay, Francesco Fedele. Geometric numerical schemes for the KdV equation. Computational Mathematics and Mathematical Physics, 2013, 53 (2), pp.221-236. ⟨10.1134/S0965542513020103⟩. ⟨hal-00694896⟩