A new type of systematic approach to study the incompressible Euler equations numerically via the vanishing viscosity limit is proposed in this work. We show the new strategy is unconditionally stable that the $L^2$-energy dissipates and $H^s$-norm is uniformly bounded in time without any restriction on the time step. Moreover, first-order convergence of the proposed method is established including both low regularity and high regularity error estimates. The proposed method is extended to full discretization with a newly developed iterative Fourier spectral scheme. Another main contributions of this work is to propose a new integration by parts technique to lower the regularity requirement from $H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best knowledge, this is one of the very first work to study incompressible Euler equations by designing stable numerical schemes via the inviscid limit with rigorous analysis. Furthermore, we will present both low and high regularity errors from numerical experiments and demonstrate the dynamics in several benchmark examples.

翻译：本文提出了一种通过消失粘性极限数值研究不可压缩欧拉方程的新型系统方法。我们证明了该新策略具有无条件稳定性：$L^2$能量耗散且$H^s$范数在时间上一致有界，无需任何时间步长限制。此外，本文建立了所提方法的一阶收敛性，包括低正则性和高正则性误差估计。该方法通过新开发的迭代傅里叶谱格式扩展至全离散情形。本工作的另一个主要贡献是提出了一种新的分部积分技术，将$L^2$误差估计所需的正则性要求从$H^4$降低至$H^3$。据我们所知，这是最早通过设计基于无粘极限的稳定数值格式并辅以严格分析来研究不可压缩欧拉方程的工作之一。此外，我们将通过数值实验展示低正则性与高正则性误差，并在若干基准算例中演示其动力学行为。