A TVD Lax-Wendroff scheme solves the Boussinesq-type equations is presented, extensively validated, and clearly demonstrated to be a robust and efficient engineering tool to simulate the physical processes involved in the tsunami wave runup and interaction of the propagating solitary waves with the idealized coastal beaches. To better understand the physical processes of the tsunami wave runup at surf zone, a parametric study concerning N-wave runup is carried out. For all cases investigated, the qualitative features of the propagating N-waves remain unaltered, even for the large wave events. The relative maximum runup height and wave steepness are found to be strongly correlated and appeared to be linearly asymptotic in form. Also, the severity of extreme wave attack is found to be a function of beach slope for a given extreme event. The numerical simulations reveal the significance of the nonlinear wave effects on the predicted maximum N-wave runup heights, which provide guidance in selecting the design height of coastal defence structures and specifying the clearance distance between the shoreline and infrastructure.

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