Approximation of a dispersive model using the Hyperbolic formulation framework: Example of the Conduit equation
Speaker 
Speaker: Boniface Nkonga
(JAD, CNRS, INRIA Sophia Antipolis)


When 
Sep 20, 2023
from 02:00 PM to 03:00 PM 
Where  Ground Floor Auditorium (006) (Hybrid) 
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Abstract:Recent papers propose hyperbolic regularizations of dispersive equations that are the EulerLagrange equations for a given Lagrangian [1]. The primary purpose is to reformulate the highorder derivative dispersive equations as a firstorder hyperbolic system. Introducing an appropriate set of parameters gives rise to an extended Lagrangian, asymptotically recovering the proper variational structure of the initial equation under concern. In the case of SerreGreenNaghdi equations, this "extended Lagrangian" method is now mathematically justified [2]. The advantage of such an approach is obvious: one can use for dispersive equations the entire arsenal of finite volume methods developed for hyperbolic equations.
We aim to apply a similar strategy but in the context of the dispersive conduit equation [3]. According to the classical formulations, this equation does not derive from a Lagrangian, and the classical variational structure is useless. In this context, the purpose is to derive a helpful mathematical pattern for the hyperbolic approximation of the conduit equation.
This presentation will exhibit a mathematical pattern preserved by the hyperbolic reformulation of the conduit equation. Other hyperbolic reformulations exist without fulfilling the proposed mathematical properties. We will compare the numerical solutions based either on hyperbolic or the dispersive formulation of the conduit equation. Contrary to the other hyperbolic models, the "structurepreserving" hyperbolic approximation shows excellent convergence results.
 N. Favrie and S. Gavrilyuk. A rapid numerical method for solving SerreGreenNaghdi equations describing long freesurface gravity waves. Nonlinearity, 2017.
 V. Duchene. Rigorous justification of the FavrieGavrilyuk approximation to the SerreGreenNaghdi model. Nonlinearity, 2019.
 N.K. Lowman and M.A. Hoefer. Dispersive hydrodynamics in viscous fluid conduits. Phys. Rev. E, 2013.
 S. Gavrilyuk, B. Nkonga, K.M. Shyue, and L. Truskinovsky. Stationary shocklike transition fronts in dispersive systems. Nonlinearity, 2020.