使用伪谱方法计算近奇异解
Computing Nearly Singular Solutions Using Pseudo-Spectral Methods
AI总结：
本文研究伪谱方法在计算流体动力学方程近奇异解时的性能，发现高阶傅里叶平滑方法比传统的2/3去混叠规则更有效，能捕获更多有效傅里叶模式且误差局部化。
摘要：
本文研究了伪谱方法在计算流体动力学方程近奇异解时的性能。我们考虑了两种消除伪谱方法中混叠误差的方式。第一种是传统的2/3去混叠规则。第二种是高阶（36阶）傅里叶平滑，它在傅里叶谱中保留了超出2/3去混叠方法截止点的大量傅里叶模式。我们考虑了一维Burgers方程和三维不可压缩Euler方程。我们证明了采用高阶傅里叶平滑的伪谱方法比采用2/3去混叠规则的伪谱方法性能更好。此外，我们表明高阶傅里叶平滑方法在每个维度上比2/3去混叠方法多捕获约12~15%的有效傅里叶模式。对于三维Euler方程，高阶傅里叶平滑方法相对于2/3去混叠方法在有效傅里叶模式上的增益可高达20%。另一个有趣的观察是，高阶傅里叶平滑方法产生的误差高度集中在解最奇异区域附近，而2/3去混叠方法倾向于在整个域中产生振荡。高阶傅里叶平滑方法在动态上也非常稳定，未观察到高频不稳定性。
英文摘要：
In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about $12 \sim 15%$ more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular, while the 2/3 dealiasing method tends to produce oscillations in the entire domain. The high order Fourier smoothing method is also found be very stable dynamically. No high frequency instability has been observed.
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