The verification cases so far considered solely the cross-shore dimension and assumed a longshore uniform coast. In the following cases the potential of the model to predict coastal and dune erosion in situations that include the 2 horizontal dimensions is further examined. A first step towards a 2DH response is to verify that the 2DH forcing by surge run-up and run-down is accurately modelled. This accuracy is controlled by the flooding and drying criterion. This criterion is tested against a numerical solution for the runup of a solitary wave on a concave bay with a sloping bottom (Figure 1) that was obtained by Ozkan \& Kirby (1997), who used a Fourier-Chebyshev Collocation model of the nonlinear non-dispersive shallow water equations. This solution is used rather than the original simulation by Zelt (1986), who used a fully Lagrangian finite element model of the shallow water equations, which included some dissipative and dispersive terms which presently cannot be modelled in XBeach. The XBeach model is run without friction, short wave forcing or diffusion.
Figure 1 Definition sketch of the concave beach bathymetry (courtesy H.T. Ozkan-Haller)
Figure 1 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 hs/Ly = 4/(10*pi), the bathymetry is given by
formula
where hs is the shelf depth, Ls is the length of the shelf in the modeled domain and Ly is the length scale of the longshore variation of the beach. This results in a beach slope hx = 0.1 in the center of the bay and of hx = 0.2 normal to the ``headlands''. In the following we chose Ly = 8 m, which determines hs = 1.0182 m. We set Ls = Ly and different values for Ls  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 runup (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
formula
which is similar to Zelt (1986)'s Eq. (5.3.7) except that we neglect the arbitrary phase shift. The phase shift to 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 a. We will compare our model to Zelt's case of a = Hs/h = 0.02 where H is the offshore wave height. Zelt found that the wave broke for a value of a = 0.03, so the present test should involve no breaking, but has a large enough nonlinearity to exhibit a pronounce two-dimensional runup. 
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*Ls we specify a no-flux (wall) boundary condition following Zelt. The numerical parameters are dx = dy = 0.125 m with a Courant number of 0.7. 
Figure 2 Comparison of present model (solid) to  Ozkan \& Kirby (1997)  (dashed): (a) Timeseries of runup in 5 stations from top to bottom y/Ly = 1, 0.75, 0.5, 0.25 and 0 respectively, and (b) Maximum runup and rundown.
Figure 2 shows the vertical runup  normalized with the offshore wave height H as a function of time, which is normalized by at the 5 cross-sections indicated in Figure 1. The solid lines represent the present model results while the dashed lines denotes Ozkan \& Kirby (1997)'s numerical results. We see that the agreement is generally good, except that the present model does not capture the second peak in the time series at y/Ly = 1 very well. This secondary peak or 'ringing' is due to the wave energy that is trapped along the coast and propagates towards the midpoint of the bay (Zelt, 1986). It is suspected that this focusing mechanism is not properly captured, because the present method approximates the shoreline as a staircase pattern, which in effect lengthens the shoreline. Also, the spatial derivatives are not evaluated parallel and perpendicular to the actual shoreline but in the fixed x and y directions. The agreement at the locations y/Ly = 0.25, y/Ly = 0 0.5 and y/Ly = 0.75 is generally good despite the large gradient of the local shoreline relative to our grid. The statistical overall score for the time series is r2 = 0.986, SCI = 0.170 and the relative bias = 0.009.
Note that Ozkan \& Kirby (1997) use a moving, adapting grid with a fixed dy (which is equal to the present model's dy in this comparison) but with a spatially and temporally varying dx so that the grid spacing near the shoreline is very small. In the present model dx is set equal to dy, which means that we can expect to have less resolution at the shoreline than Ozkan \& Kirby (1997).
Figure 15b shows the maximum vertical runup and rundown, normalized by H, versus the alongshore coordinate y. It is seen that the maximum runup agrees well with Ozkan \& Kirby (1997) but that the maximum rundown is not represented well in the center of the domain. The ‘wiggles’ in the solid line are evidence of the staircasing of the shoreline: since the shoreline is not treated as a continuous but rather as a discrete function, so is the runup in the individual nodes. The statistical score for the maximum run-up is r2 = 0.98, SCI= 0.04 and the relative bias = -0.03. 
The above results are consistent with the results obtained with the SHORECIRC model which is based on similar hydrodynamic equations, see Van Dongeren and Svendsen (1997b), and show that also the current model is capable of representing run-up and run-down.