Approximation of a dispersive model using the Hyperbolic formulation framework: Example of the Conduit equation

Abstract:Recent papers propose hyperbolic regularizations of dispersive equations that are the Euler-Lagrange equations for a given Lagrangian [1]. The primary purpose is to reformulate the high-order derivative dispersive equations as a first-order 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 Serre-Green-Naghdi 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 "structure-preserving" hyperbolic approximation shows excellent convergence results.

1. N. Favrie and S. Gavrilyuk. A rapid numerical method for solving Serre-Green-Naghdi equations describing long free-surface gravity waves. Nonlinearity, 2017.
2. V. Duchene. Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model. Nonlinearity, 2019.
3. N.K. Lowman and M.A. Hoefer. Dispersive hydrodynamics in viscous fluid conduits. Phys. Rev. E, 2013.
4. S. Gavrilyuk, B. Nkonga, K.M. Shyue, and L. Truskinovsky. Stationary shock-like transition fronts in dispersive systems. Nonlinearity, 2020.