Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$2/3 de-aliasing method
Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$2/3 de-aliasing method
复制标题
非线性问题傅里叶方法的稳定性和谱收敛性：浅谈$$2/3$$2/3去锯齿方法的缺点
DOI：
10.1007/s00211-014-0652-y
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发表时间：
2013
影响因子：
2.1
通讯作者：
E. Tadmor
中科院分区：
文献类型：
--
作者：
C. Bardos;E. Tadmor
傅里叶方法的高阶精度使其成为许多大规模模拟的首选方法。我们在这里讨论非线性演化问题的傅里叶方法的稳定性，重点讨论无粘伯格斯方程和多维不可压缩欧拉方程这两个典型情况。用于处理此类二次非线性问题的傅立叶方法主要有两种类型。一种是谱傅里叶方法。另一种是 $$2/3$$2/3 伪频谱傅里叶方法，其中删除频谱的最高 $$1/3$$1/3 部分；这通常是维持二次能量平衡和避免混叠误差的选择方法。本文讨论了两个主题。首先，我们证明只要底层精确解具有最小的 $$C^{1+\alpha }$$C1+α 空间规律性，那么谱和 $$2/3$$2/3 伪谱傅立叶方法都是稳定的。因此，我们证明了它们对于无粘 Burgers 方程和不可压缩 Euler 方程的平滑解的谱收敛性。另一方面，我们证明，在基础解缺乏足够平滑度的关键时间之后，谱和 $$2/3$$2/3 伪谱傅里叶方法都表现出通过寄生振荡实现的非线性不稳定性。特别是，在无粘 Burgers 方程中激波形成后，有界（伪）谱傅里叶解的总变化必须随着模式数量的增加而增加，我们规定 3D 不可压缩欧拉方程也会出现类似的情况：极限傅里叶解被证明强制 $$L^2$$L2 能量守恒，并且与能量耗散 Onsager 解的对比通过寄生振荡反映出来。
The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers’ equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the $$2/3$$2/3 pseudo-spectral Fourier method, where one removes the highest $$1/3$$1/3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal $$C^{1+\alpha }$$C1+α spatial regularity, then both the spectral and the $$2/3$$2/3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the $$2/3$$2/3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers’ equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce $$L^2$$L2-energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.
影响因子：
2.5
作者：
De Lellis, Camillo;Szekelyhidi, Laszlo, Jr.
通讯作者：
Szekelyhidi, Laszlo, Jr.