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 $\alpha$-耗散解的一种收敛数值方法。该方法首先对初始数据施加定制投影算子，然后利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环节，因此其设计至关重要：既要确保初始数据的良好逼近，又要保证后续时刻能量耗散误差保持较小。进一步证明，波形这一主要关注量在所有 $t \geq 0$ 时刻均在 $L^{\infty}$ 意义下收敛，而能量密度的子序列在几乎所有时刻弱收敛。