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 $T \geq 0$.

翻译：本文针对Hunter-Saxton方程的α-耗散解提出了一种新的数值方法，其中α属于$W^{1, \infty}(\mathbb{R}, [0, 1))$。该方法将投影算子、广义特征线法以及一种迭代格式相结合，该迭代格式基于在断裂时间聚集时强制施加最小时间步长。数值算例表明，这些最小时间步长显著提高了算法的效率。此外，证明了对于任意有限时间$T \geq 0$，波形在$C([0, T], L^{\infty}(\mathbb{R}))$空间中收敛。