  We propose a new numerical method for $\alpha$-dissipative solutions of the
Hunter-Saxton equation, where $\alpha$ belongs to $W^{1, \infty}(\mathbb{R},
[0, 1))$. The method combines a projection operator with a generalized method
of characteristics and an iteration scheme, which is based on enforcing minimal
time steps whenever breaking times cluster. Numerical examples illustrate that
these minimal time steps increase the efficiency of the algorithm
substantially. Moreover, convergence of the wave profile is shown in $C([0, T],
L^{\infty}(\mathbb{R}))$ for any finite $T \geq 0$.
