Figure - numerical solutions based on piecewise self-similar functions.

Figure 1 shows a approximation for the initial data $u(x,0)=\sin(\pi x), 0\le x\le1$ and Figure 2 displays the solution at $t=1.2$. From them we may observe that the limit of the solution always passes through the artificial shocks and the size of artificial shocks decreases in time. This property provides good estimates. Since this computation does not require time discretization, it saves computation time, and hence for multi-dimensional problems it can be helpful to save computational time