Gaël Richard - Hyperbolic approximation of the Serre-Green-Naghdi equations

Hyperbolic approximation of the Serre-Green-Naghdi equations and extension to weakly compressible flows

The Serre-Green-Naghdi equations are fully nonlinear and weakly dispersive equations for shallow-water flows. They are commonly used to model coastal waves. However the numerical resolution of these equations is difficult and computationally expensive because an elliptic step has to be solved at each time step. This problem results from the assumption of incompressibility which creates a non-local effect since pressure variations propagate at an infinite celerity in an incompressible fluid. Hyperbolic approximations of the Serre-Green-Naghdi equations have been recently derived. These approaches are related to the general method of taking into account the compressibility of the fluid and the propagation of acoustic waves at a finite speed. Furthermore static compressibility can have measurable effects in the case of tsunamis propagating in deep oceans. The derivation of a hyperbolic approximation of the Serre-Green-Naghdi equations is presented. This model can be extended to weakly compressible fluids.