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

Abstract

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.

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

“…2). We note that similar inequalities for the phase velocities were obtained in [8] for one-layer flows (η 0 = 0). The linear analysis allows us to estimate the values of the relaxation parameter α to achieve the required approximation accuracy using model (12).…”

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

“…Here, q i might be scalars or entries of a vector or tensor field, f i (q) are sufficiently smooth functions of the entire vector of state variables. For example, equations (18a), (18a) and ordinary differential equations of type (24) constitute the nonlinear elastoplasticity model studied in [31]. The complimentary structure of the SHTC equations can be also seen from the point of view of the odd-even parity with respect to time-reversal transformation (TRT) [71,74] which has been playing a central role in thermodynamics for very long time.…”

Entropy‐satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

“…Now, we focus on the dispersion step (NH .b)-(NH .c). As explained above, the dispersion step (NH .b) is equivalent to (10). By multiplying the second Equation of (10) by u n+1 k , it yields…”

“…For example, in the work of Le Métayer et al, Lagrangian variables are used. In the work of Favrie and Gavrilyuk, a relaxation of the constrains are used to obtained a hyperbolic system. However, the limit of the relaxation parameters to obtain the original Green‐Naghdi model is not clear in the numerical framework.…”