Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: Applications to the Serre-Green-Naghdi equations

In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing shallow water flows, and Iordanskii-Kogarko-Wijngaarden (IKW) fluids containing small compressible gas bubbles. These models are Euler-Lagrange for a given Lagrangian share common mathematical structure, namely dependence of pressure on material derivatives macroscopic variables. We develop generic model such that SGN IKW systems become its special cases if only one specifies appropriate Lagrangian, then use extended Lagragian approach proposed in Favrie Gavrilyuk (2017) to build hyperbolic approximation. The new approximate is unconditionally both cases, accurately describes phenomena, which allows impose discontinuous initial data shock waves. consider 2-D version system as an example numerical simulations apply second order implicit-explicit scheme numerically integrate system. obtained 1-D results close agreement with available exact solutions tests. \textbf{Keywords: } equations, bubbly fluids, conservation laws, waves, methods