  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 on
several important practical examples. Finally, the simulation results are
compared with the well-known analytical and experimental predictions.
