\autoref{fig:zeltbath.png} shows the definition sketch of the concave beach bathymetry in the present coordinate system, converted from the original system by Zelt (1986). The bathymetry consists of a flat bottom part and a beach part with a sinusoidally varying slope. For Zelt (1986)'s fixed parameter choice of $\sqrt{\beta}\,=\,\frac{h_s}{L_y} = \frac{4}{10\pi}$, the bathymetry is given by
\begin{eqnarray} h =\, \left\{\begin{array}{lcl} \\ h_s & \mbox{,} & x \leq L_s \\ h_s - \frac{0.4\,(x\,-\,L_s)}{3\,-\,\cos\left( \frac{\pi y}{L_y} \right)} & \mbox{,} & x > L_s \\ \end{array} \right. \label{3.36} \end{eqnarray}
\noindent where $h_s$ is the shelf depth, $L_s$ is the length of the shelf in the modeled domain and $L_y$ is the length scale of the longshore variation of the beach. This results in a beach slope of $h_x = \frac{1}{10}$ in the center of the bay and of $h_x = \frac{1}{5}$ normal to the ``headlands''. In the following we chose $L_y = 8\,m$, which determines $h_s = 1.0182 \, m$. We set $L_s = L_y$. Different values for $L_s$ only cause  phase shifts in the results, but no qualitative difference, so this parameter is not important in this problem. Also indicated in the figure are the five stations where the vertical run-up (the surface elevation at the shoreline) will be measured. 
At the offshore ($x=0$) boundary we specify an incoming solitary wave, which in dimensional form reads
\begin{eqnarray} {\zeta}_i\,(t)\,=\,\alpha\,\,h_s\,\mbox{sech}^2\left( \sqrt{\frac{3\,g}{4\,h_s}\,\alpha\,(1+\alpha)}\;\;\;(t\,-\,t_o)\,\right) \label{3.37} \end{eqnarray}
which is similar to Zelt (1986)'s Eq. (5.3.7). The phase shift $t_o$ is chosen such that the surface elevation of the solitary wave at $t=0$ is 1\% of the maximum amplitude. The only parameter yet to be chosen is $\alpha$. We will compare our model to Zelt's case of $\alpha\,=\,\frac{H}{h_s}\,=\,\,0.02$, where $H$ is the offshore wave height. Zelt found that the wave broke for a value of $\alpha\,=\,0.03$, so the present test should involve no breaking, but has a large enough nonlinearity to exhibit a pronounced two-dimensional run-up. 
Any outgoing waves will be absorbed at the offshore boundary by the absorbing-generating boundary condition. At the lateral boundaries $y=0$ and $y=2\,L_y$ we specify a no-flux (wall) boundary condition following Zelt. The model equations used in this test are the nonlinear shallow water equations without forcing or friction. The numerical parameters are $\Delta x\,=\,\Delta y\,=\frac{1}{8}\,m$ with a Courant number $\nu\,=\,0.7$.