Hamiltonian Approach to Fairly Low and Fairly Long Gravity Waves

摘要

The study relates to irrotational nonlinear dispersive gravity waves in aninviscid incompressible fluid. In Chapter 2 we derive an alternative positive-definite first-order approximation to the Hamiltonian. In the implementation of this alternative only two Helmholtz-type equations need to be solved and the matrix is symmetric. Numerical aspects are considered in Chapter 3. For the time-integration of our Boussinesq equations the classical fourth-order Runge-Kutta method was used and for spatial discretization we used fourth-order accurate discretization schemes. The conjugate gradient method was employed in conjunction with a preconditioning for solving the Helmholtz-type equations. Two preconditioning techniques were used: polynomial preconditioning and preconditioners based on incomplete Cholesky decomposition. Some first results are presented in Chapter 4. The results obtained by the two models differ only by a few percents, which is quite acceptable. The speed improvement is about 2.5.