Numerical shallow water wave modeling

The mild-slope equation, or the combined refraction-diffraction equation, finds wide application in coastal and ocean engineering. In this thesis, numerical models were developed and various novel numerical techniques were investigated for practical applications. These models were verified against analytical solutions and experimental data at every stage. Simulations of harbor/coastal waves were also performed.; Because the mild-slope equation is an elliptic boundary value problem, the solution has to be obtained simultaneously for the entire domain, which often presents a very large system of linear equations. Therefore the solution technique is extremely crucial when the model is used in a large domain. An iterative scheme would be the best choice for this purpose. Because of the nature of the mild-slope equation, the final system of equations does not converge with conventional iterative schemes (such as Jacobi, Gauss-Seidel, SOR, etc.). Here we have developed a special procedure to transform the "original" system into a system that guarantees convergence and then adopted the conjugate gradient iterative technique to solve the linear system of equations.; Boundary element, finite-difference and finite-element methods to be used in solving the mild-slope equation were studied for various applications. For constant water depth, the mild-slope equation reduces to the Helmholtz equation (the so called diffraction equation), which can be easily solved by the boundary element method. For arbitrary bathymetries, however, the mild-slope equation must be solved by either the finite-difference or the finite-element method. For problems with simple geometries, the finite-difference method is attractive because it is much easier to construct the numerical grid and to code than finite-element models. In general, though, the topography of a coastal region is very complex and the finite-element method is a better choice, since it can better represent the actual shape of the domain and allows flexibility in the construction of elements. A complete finite-element shallow water wave model package (CGWAVE) is developed and presented in this thesis.; The traditional boundary treatment in elliptic shallow water models often results in erroneous oscillations in the model domain. Several new techniques which avoid these defects of the traditional method were developed. This enables us to tackle real harbor problems with much better results. These new techniques were also combined in CGWAVE, which can be used in practical large-domain problems with complex geometries.