Solitary Wave Generation in Shallow Water Regions Using High Order Numerical Piston Paddle

Water wave theory has been studied for centuries to understand various physical phenomena in the estuary or coastal areas. In particular, research on the solitary wave is a classical topic, which has been studied over the past 100 years since John Scot Russell`s remarkable observation in 1834. Understanding about this topic is very important for various issues in the engineering fields, such as tsunami, inundation, wave run-up and overtopping. Meanwhile, due to the advent of computers since 1970, numerical studies have been actively carried out so far, taking over the baton of past studies. Thanks to the significant improvement of the computation power and resource, numerical studies that solve the partial differential equations representing physical phenomena more precisely and accurately are drawing much attention. In accordance with the trend, this study deals with how to generate solitary waves accurately with a high order numerical method. For comparison with the results of past experimental studies, piston type wave-maker is implemented through the fifth order weighted essentially non-oscillatory (WENO) method, and which is governed by the conservative form of nonlinear shallow water equation. Numerical tests are performed, which are based on the Boussinesq`s solitary waves (Boussinesq, 1871; Rayleigh, 1876; Goring, 1978). Numerical results show that if the ratio of amplitude to water depth is large, waves with inadmissible errors are generated. These errors occur due to the violation of a hydrostatic pressure assumption, which are mainly used in shallow water equations, especially when the variations of the terrain or water surface level are relatively large compared to water depth. In order to avoid contamination from these errors and increase the accuracy of numerical experiments, this study provides a gravity correction method (GCM) reflecting the non-hydrostatic pressure explicitly. The GCM allows more accurate solitary waves to be generated beyond the limit of hydrostatic shallow water equation.