A conservative finite element method for the Korteweg-de Vries equation

Abstract

A finite element method for the 1-periodic Korteweg-de Vries equation \[ u t + 2 u u x + u x x x = 0 {u_t} + 2u{u_x} + {u_{xxx}} = 0 \] is analyzed. We consider first a semidiscrete method (i.e., discretization only in the space variable), and then we analyze some unconditionally stable fully discrete methods. In a special case, the fully discrete methods reduce to twelve point finite difference schemes (three time levels) which have second order accuracy both in the space and time variable.