  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.
