Generalized Riemann problem for dispersive equations

Abstract : We study the inertia type regularization of equations for barotropic fluids. This regularization is formally obtained as the Euler-Lagrange equations for a Lagrangian containing terms which are quadratic with respect to the material derivative of density. Such a regularization arises, in particular, in the modeling of waves in bubbly fluids as well as in the theory of water waves (Serre-Green-Naghdi equations). We show that such terms are not always regularizing. The solution can develop shocks even in the presence of dispersive terms. In particular, we found such a shock solution relating a constant state with a periodic wave train. The shock speed coincides necessarily with the velocity of the corresponding wave train. The associated Rankine-Hugoniot relations correspond to Whitham's equations (modulation equations) of the regularized system. The numerical evidence of the existence of such shocks is demonstrated. To this aim, a robust high-order accurate numerical has been designed based on an appropriate operator splitting of the governing equations. In particular, it has been shown that such waves can dynamically be formed. Also, when such a wave is used for initial data, it is not destroyed by small perturbations. This proves a certain stability of these waves.
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Submitted on : Monday, December 17, 2018 - 10:39:26 PM
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Sergey Gavrilyuk, Boniface Nkonga, Keh-Ming Shyue, Lev Truskinovsky. Generalized Riemann problem for dispersive equations. 2018. ⟨hal-01958328⟩

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