Numerical integration of the shallow water equations over a sloping shelf

Abstract

SUMMARYA finite-difference method is described for the numerical integration of the one-dimensional shallow water equations over a sloping shelf that allows for a continuously moving shoreline. An application of the technique is made to the propagation of non-breaking waves towards the shoreline. The results of the computation are compared with an evaluation based upon an exact analytical treatment of the non-linear equations. Excellent agreement is found for both tsunami and tidal scale oscillations.

A particle finite element method applied to long wave run-up

“…A related approach is the use of the arbitrary Lagrangian Eulerian (ALE) methods where the computational domain is mapped onto a ÿxed coordinate region, generally consisting of one or more rectangles, but where the nodes not necessarily correspond to particles. Examples of solutions of the shallow water equations by ALE techniques are found in References [28,[35][36][37][38]. The last of these is a brief paper where a meshless technique, utilizing smooth shape functions and collocation is applied to plane waves.…”

Validation of ANUGA hydraulic model using exact solutions to shallow water wave problems

“…Note that this analytical solution is dimensionless, where the original dimensional quantities have been scaled [2,9,14]. The scalings are as follow:…”

Approximations of the Carrier–Greenspan periodic solution to the shallow water wave equations for flows on a sloping beach

“…Our main point is that, in general, the approximate solution of Johns is unable to accurately estimate the Carrier–Greenspan exact solution at the zero point of the spatial domain. In addition, the large prescription error at the fixed boundary results in a large error of the solution generated by the numerical method.…”

“…The Carrier–Greenspan periodic solution has been widely applied to test the performance of numerical methods used to solve the shallow water wave equations (consult for example). This solution involves a fixed boundary and a moving boundary.…”

“…Johns presented a specific form of the Carrier–Greenspan exact solution of periodic type over a spatial domain and prescribed an approximate solution at the zero point of the spatial domain, where the fixed boundary is located. The approximate solution can be applied at the fixed boundary to generate an approximation of the Carrier–Greenspan exact solution using a numerical method as done by a number of authors, such as Johns and Sidén and Lynch . Thus, the numerical solution is affected by two sources of error, namely, the error in the boundary condition (at the fixed boundary) and the error introduced by the numerical discretization.…”