A new variational (dis)continuous Galerkin finite element method is presented for the linear free surface gravity water wave equations. We formulate the space-time finite element discretization based on a variational formulation analogous to Luke's variational principle. The linear algebraic system of equations resulting from the finite element discretization is symmetric with a very compact stencil. To build and solve these equations, we have employed PETSc package in which a block sparse matrix storage routine is used to build the matrix and an efficient conjugate gradient solver is used to solve the equations. The numerical scheme is verified for linear harmonic free surface waves in a periodic domain and linear free surface generated by a harmonic wave maker in a rectangular wave basin. We conclude that the scheme is second order accurate and, shows no dissipation and minimal dispersion errors in the wave propagation.
|Publisher||Department of Applied Mathematics, University of Twente|