Summary: A linearized implicit finite difference method is devised for $K(n,n)$. The stability and accuracy of the proposed methods are discussed. A compacton wave solution of the equation $K(n,n)$ is used to examine the accuracy and efficiency of the proposed methods and study the effect of the added artificial dissipation term to solve the $K(n,n)$ equation using finite difference method. The dynamics of waves having various initial wavepackets are discussed.