A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

A new numerical method for solving the Serre–Green–Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler–Lagrange equations for a 'master' lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the 'master' lagrangian by a one-parameter family of 'augmented' lagrangians, depending on a greater number of variables, for which the corresponding Euler–Lagrange equations are hyperbolic. In such an approach, the 'master' lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of 'Favre waves' representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.