Boussinesq modelling of solitary wave and N-wave runup on coast

A total variation diminishing Lax–Wendroff scheme has been applied to numerically solve the Boussinesq-type equations. The runup processes on a vertical wall and on a uniform slope by various waves, including solitary waves, leading-depression N-waves and leading-elevation N-waves, have been investigated using the developed numerical model. The results agree well with the runup laws derived analytically by other researchers for non-breaking waves. The predictions with respect to breaking solitary waves generally follow the empirical runup relationship established from laboratory experiments, although some degree of over-prediction on the runup heights has been manifested. Such an over-prediction can be attributed to the exaggeration of the short waves in the front of the breaking waves. The study revealed that the leading-depression N-wave produced a higher runup than the solitary wave of the same amplitude, whereas the leading-elevation N-wave produced a slightly lower runup than the solitary wave of the same amplitude. For the runup on a vertical wall, this trend becomes prominent when the wave height-to-depth ratio exceeds 0.01. For the runup on a slope, this trend is prominent before the strong wave breaking occurs.