From the numerical viewpoint in the simulation of wave run-up, most of the problems arise in the determination of the shoreline position since the shoreline moves up and down the sloping beach during the run-up and run-down processes. In other words, investigation of long wave run-up can be categorized as a moving boundary problem, and consequently, the main question to be answered is how to track this boundary. Therefore, a special treatment is necessary in the numerical model to define the shoreline position. In the present study, by recourse to a Lagrangian-Eulerian transformation, the time-varying physical domain is converted to a fixed-length time-invariant computational domain. However, this computational mapping introduces additional non-linear terms into the governing equations. By imposing appropriate boundary conditions on the shoreline, the mapped equations are then solved using the two-step Richtmyer-Lax-Wenderoff numerical scheme. All of the computations are carried out by employing a program code developed by authors in FORTRAN 90. The simulated surface profiles show very satisfactory agreement with available analytical solutions as well as the experimental data.
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