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Hunter--Saxton方程$α$-耗散解的一种收敛数值方法。该方法基于对初始数据施加量身定制的投影算子，然后使用广义特征方法精确求解。投影步骤是唯一引入近似误差的步骤。因此，其设计至关重要，不仅要确保初始数据得到良好近似，还要保证后续时刻由能量耗散导致的误差保持较小。此外，研究证明，波剖面这一主要关注量在$t \geq 0$时在$L^{\infty}$意义下收敛，而能量密度的子序列在几乎所有时刻弱收敛。