Numerical Simulation for One-Dimensional (1D) Wave Propagation by Solving the Shallow Water Equations using the Preissmann Implicit Scheme

Abstract

This research simulated one-dimensional wave propagation by solving the shallow water equations using the Preissman implicit numerical scheme due to its ability to maintain simplicity and stability at a larger time step value. This numerical model was fundamentally developed to satisfy the shallow water condition, where the water depth or horizontal-length scale is much smaller than the free-surface disturbance wavelength or vertical-length scale, and to comprehensively test the accuracy of the model. Consequently, three different types of waves were considered and these include (1) tidal, (2) roll, and (3) solitary. In the first case, the model was proven to be robust and accurate due to its relatively-small errors for both water-surface elevation and velocity indicating that the Preismann scheme is suitable for longwave simulations. In the second case, it was fairly accurate in capturing the periodic permanent roll waves despite showing a higher water-surface elevation than the one observed and this discrepancy is due to the neglect of the turbulent Reynold stress in the model. Meanwhile, the last case showed remarkable discrepancies in the water-surface elevation because the dispersion effect is quite significant during the wave propagation. This indicates that the Preismann scheme underestimated the wave crest along with time when the dispersion term was neglected. All simulations were performed using the tridiagonal matrix algorithm, thereby eliminating the need for iterations for the solution of the Preismann scheme. The findings of this study are beneficial to the next generation of the Preissmann-scheme models which can be designed to include turbulence and dispersion terms.