Error estimates for Galerkin approximations of the Serre equations

TL;DR

This paper provides optimal error estimates for Galerkin finite element approximations of the Serre equations, a nonlinear dispersive system modeling long wave propagation, and demonstrates the scheme's effectiveness through numerical experiments.

Contribution

It establishes the first optimal-order $L^{2}$-error estimates for Galerkin discretizations of the Serre equations and applies high-order time-stepping for accurate simulations.

Findings

Optimal $L^{2}$-error bounds proven for Galerkin method

Numerical simulations accurately capture solitary wave phenomena

High-order schemes effectively resolve wave interactions

Abstract

We consider the Serre system of equations which is 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.