标题： 在存在不连续性情况下的谱方法 显示英文标题

标题： Spectral Methods in the Presence of Discontinuities

摘要： 谱方法为数值求解各种微分方程提供了一种优雅而有效的方法。 对于平滑问题，谱方法的截断误差在无穷范数和$L_2$-范数下呈指数级消失。 然而，对于非平滑问题，收敛性显著变差——对于不连续问题，误差的$L_2$-范数将以次线性速率收敛，而无穷范数则根本不会收敛。 我们探索并改进了一种后处理技术——最优收敛的平滑函数——以从非平滑数据的收敛性较差的谱重构中恢复指数收敛性。 这是将这些技术用于现实系统模拟的重要第一步。 显示英文摘要

摘要： Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and $L_2$-norm. However, for non-smooth problems, convergence is significantly worse---the $L_2$-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique---optimally convergent mollifiers---to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. This is an important first step towards using these techniques for simulations of realistic systems.