修正的Hunter-Saxton方程保守有限差分方案的稳定性与收敛性
Stability and convergence of a conservative finite difference scheme for the modified Hunter--Saxton equation
AI总结：
本文研究修正的Hunter-Saxton方程的数值解法，提出保守有限差分方案，证明其在统一范数下的稳定性及均匀收敛性，利用离散守恒律克服混合导数带来的困难。
摘要：
修正的Hunter-Saxton方程用于建模短波的传播。由于其混合导数，其在周期域上的初值问题比标准演化方程复杂得多。尽管其局部well-posed性已近期被证明，但其解的行为仍待研究。本文为解决此问题，推导出保守有限差分方案，并严格证明其在统一范数下的稳定性及均匀收敛到足够光滑的精确解。利用离散守恒律克服混合导数带来的困难。
英文摘要：
The modified Hunter--Saxton equation models the propagation of short capillary-gravity waves. As it involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary equations. Although its local well-posedness has recently been proved, the behavior of its solution is yet to be investigated. In this paper, to develop a reliable numerical method for this problem, we derive a conservative finite difference scheme. Then, we rigorously prove not only its stability in the sense of the uniform norm but also its uniform convergence to sufficiently smooth exact solutions. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative.
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