  We compare the numerical solutions of the 2+1 equivariant Wave Map problem
computed with the symplectic, constraint respecting Rattle algorithm and the
well known fourth order Runge-Kutta method. We show the advantages of the
Rattle algorithm for constrained system compared to the free evolution with the
Runge-Kutta method. We also present an expression, which represents the energy
loss due to constraint violation. Taking this expression into account we can
achieve energy conservation for the Runge-Kutta scheme, which is better than
with the Rattle method. Using the symplectic scheme with constraint enforcement
we can reproduce previous calculations of the equivariant case without imposing
the symmetry explicitly, thereby confirming that the critical behaviour is
stable.
