On the solution of the slope beach problem in the context of shallow-water code benchmarking: Why non-linearization of the initial waveforms is essential

Abstract

We detail a new semi-analytic solution for the slope beach problem in which linearization of the initial waveform is not carried out to provide the exact corresponding boundary condition for the transformed problem. Numerical code benchmarking requires the construction of exact solutions to check the convergence order and efficiently compare the simulations using fine meshes. We quantify the impact of the linearization assumption and compare two situations (linearized and non-linearized) with a second-order finite volume code. The accuracy of the shoreline velocity, maximum run-up, and maximum run-down situations, in particular, are of crucial importance in applications as they characterize the inundation and impact intensity of a wave on potential infrastructure. To this end, we provide two sets of spreadsheets, containing highly accurate approximations of the exact non-linearized solution for cases (c) and (d) of Carrier et al. (2003) study, for benchmarking purposes.