A convergent numerical method for $\alpha$-dissipative solutions of the Hunter-Saxton equation is derived. The method is based on applying a tailor-made projection operator to the initial data, and then solving exactly using the generalized method of characteristics. The projection step is the only step that introduces any approximation error. It is therefore crucial that its design ensures not only a good approximation of the initial data, but also that 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 energy density converges weakly for almost every time.

翻译：本文推导了一种求解Hunter-Saxton方程$\alpha$-耗散解的收敛数值方法。该方法首先对初始数据应用一个定制的投影算子，然后使用广义特征线法精确求解。投影步骤是引入近似误差的唯一环节。因此，其设计不仅要保证对初始数据的良好逼近，还必须确保后续时间步中由能量耗散引起的误差保持较小，这一点至关重要。此外，研究证明，主要关注量——波剖面，对所有$t \geq 0$均在$L^{\infty}$范数下收敛；而能量密度的子序列在几乎每个时刻均弱收敛。