Discrete asymptotic equations for long wave propagation (Q2834567)

The purpose of this paper is to discretize partially the incompressible Euler equations in the horizontal direction using a finite element method, and then perform a formal asymptotic analysis similar to the one used for continuous Boussinesq equations. The proofs only deal with two-dimensional and one-dimensional problems. The first theorem gives an estimate for the solution of the finite element discrete matrix equations \(\star\). Then the final new discrete numerical model plus the continuity equation are obtained by substitution from \(\star\) by line integration. Then discrete Leibniz's rule, i.e., consistency with Peregrine's model and linear dispersion characteristics are studied by Taylor expansion and introduction of new variables. Finally, the phase velocity, wave amplitude and grid convergence are plotted.