The classical FEM

The Galerkin approximation in this case is a classical approximation:

where are the classical piecewise linear shape functions. In this case, we don't need to integrate by parts. The formulation is:

where the time stepping scheme is following:

The classical FEM gives oscillations near the shock (see Fig. 15).

Figure 15: The results for the classical FEM.

We note that the oscillations propagate in time, which was foreseeable, because of the history term on the right hand side.