A new run-up algorithm based on local high-order analytic expansions

• New asymptotic solutions in the vicinity of the shoreline are derived.
• A novel run-up algorithm is proposed based on more accurate knowledge of the shoreline dynamics.
• This algorithm is tested against the known analytical solutions and experimental data.

The practically important problem of the wave run-up is studied in this article in the framework of Nonlinear Shallow Water Equations (NSWE). The main novelty consists in the usage of high order local asymptotic analytical solutions in the vicinity of the shoreline. Namely, we use the analytical techniques introduced by S.  Kovalevskaya and the analogy with the compressible gas dynamics (i.e.  gas outflow problem into the vacuum). Our run-up algorithm covers all the possible cases of the wave slope on the shoreline and it incorporates the new analytical information in order to determine the shoreline motion to higher accuracy. The application of this algorithm is illustrated in several important practical examples. Finally, the simulation results are compared with the well-known analytical and experimental predictions.