Variational Space-time (Dis)continuous Galerkin Method for Linear Free Surface Waves

A new variational (dis)continuous Galerkin finite element method is presented for linear free surface gravity water wave equations. In this method, the space-time finite element discretization is based on a discrete variational formulation analogous to a version of Luke’s variational principle. The finite element discretization results into a linear algebraic system of equations with a symmetric and compact stencil. These equations have been solved using the PETSc package, in which a block sparse matrix storage routine is used to build the matrix and an efficient conjugate gradient solver to solve the equations. The finite element scheme is verified against exact solutions: linear free surface waves in a periodic domain and ones generated by a harmonic wave maker in a rectangular wave basin. We found that the variational scheme has no dissipation and minimal dispersion errors in the wave propagation, and that the numerical results obtained are (p+1)-order accurate for a p-order polynomial approximation of the wave field.