The discretized Boussinesq equation and its conditional symmetry reduction

Abstract

In this article we show that we can carry out the symmetry preserving discretization of the Boussinesq equation with respect to three of its more significant conditional symmetries. We perform the symmetry reduction of the obtained nonlinear discrete schemes with respect to the conditional symmetries and obtain the reduced discrete equations which unlike in the continuous case, are not always reducible to second order difference equations. A numerical comparison with the exact continuous solution given by Weierstrass elliptic functions is carried out.