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:计算出猎杀- 萨克斯顿方程式的共性数字方法。 方法基于对初始数据应用一个量身定做的投影操作员, 然后精确地使用通用特性方法解决。 投影步骤是引入任何近似错误的唯一步骤。 因此, 关键在于其设计不仅确保初始数据有一个良好的近似值, 并且确保后来因能量流失造成的错误仍然很小。 此外, 显示主要利息数量, 即波形图, 以$L<unk> infty} 来对全部 $\ geq 0 进行汇合, 而能量密度的子序列几乎每次都会以微弱的汇合 。</s>