Error Estimates for Galerkin Approximations of the Serre Equations

Abstract

We consider the Serre system of equations, a nonlinear dispersive system that models two-way propagation of long waves of not necessarily small amplitude on the surface of an ideal fluid in a channel. We discretize in space the periodic initial-value problem for the system using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove an optimal-order $L^{2}$-error estimate for the resulting semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge--Kutta scheme for time stepping, we construct a highly accurate fully discrete scheme in order to approximate solutions of the system, in particular, solitary-wave solutions, and study numerically phenomena such as the resolution of general initial profiles into sequences of solitary waves, and overtaking collisions of pairs of solitary waves propagating in the same direction with different speeds.