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A convergent numerical method for $\alpha$-dissipative solutions of theHunter--Saxton equation is derived. The method is based on applying atailor-made projection operator to the initial data, and then solving exactlyusing the generalized method of characteristics. The projection step is theonly step that introduces any approximation error. It is therefore crucial thatits design ensures not only a good approximation of the initial data, but alsothat errors due to the energy dissipation at later times remain small.Furthermore, it is shown that the main quantity of interest, the wave profile,converges in $L^{\infty}$ for all $t \geq 0$, while a subsequence of the energydensity converges weakly for almost every time.

A numerical algorithm for $\alpha$-dissipative solutions of the Hunter--Saxton equation