Behaviour of the Serre equations in the presence of steep gradients revisited

Abstract

We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. Lastly the numerical solutions are used to evaluate the usefulness of asymptotic results in the literature to approximate the depth and location of the front of an undular bore.

Stationary shock-like transition fronts in dispersive systems

“…Finally, we present numerical results for the simulation of a dam break problem studied, for instance, in [12,39,48]. Since there is no analytical solution to this problem, such a study is rather qualitative, but it allows us to recover some qualitative characteristics of the solution (the amplitude of the leading wave and its velocity, for example).…”

“…where α = 2 m or α = 0.4 m. The structure of the solution (but not the velocity of the leading solitary wave and its velocity) depends on the value of α. According to the terminology given in [48], the case α = 2 m produces S 2 configuration (flat structure of the fluid depth behind the . As far as the global wave structure is concerned, our results are in good agreement with the ones shown in [12] at time t = 150 s, where a different value of the gravitational constant, g = 1 m/s 2 , was employed.…”

“…Left figure : Numerical result for the dam break problem for the initial data(63) with α = 2 m (S 2 case in the terminology of[48]). The solid line is the water depth at time t = 47.434 s, and the dashed line is the initial condition.…”

Importance of dispersion for shoaling waves

“…These undulations oscillate around the bore of the shallow water wave equations, significantly increasing its maximum amplitude. These undulations cause an undular bore to travel slightly quicker than the bores of shallow water wave equations as demonstrated by Pitt et al (2017). These results suggest that the shallow water wave equations will slightly underestimate the arrival time as well as the maximum amplitude of a steep advancing wave.…”

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

“…To approximate the intercell fluxes F n j+1∕2 (Q n ) and F n j−1∕2 (Q n ) we use the method of Kurganov et al 17 While the source term is approximated with the well-balancing modifications proposed by Audusse et al 21 To achieve a second-order accurate method using (5), second-order accurate approximations to h, u, G, u/ x, b/ x, 2 b/ x 2 inside a cell are required. The conserved quantities h and G are reconstructed from the cell averages, resulting in a linear approximation to h and G over the cell.…”