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首页> 外文期刊>Journal of Computational Physics >Variational multiscale stabilization of high-order spectral elements for the advection-diffusion equation
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Variational multiscale stabilization of high-order spectral elements for the advection-diffusion equation

机译：对流扩散方程的高阶谱元素的变分多尺度稳定

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摘要

One major issue in the accurate solution of advection-dominated problems by means of high-order methods is the ability of the solver to maintain monotonicity. This problem is critical for spectral elements, where Gibbs oscillations may pollute the solution. However, typical filter-based stabilization techniques used with spectral elements are not monotone. In this paper, residual-based stabilization methods originally derived for finite elements are constructed and applied to high-order spectral elements. In particular, we show that the use of the variational multiscale (VMS) method greatly improves the solution of the transport-diffusion equation by reducing over- and under-shoots, and can be therefore considered an alternative to filter-based schemes. We also combine these methods with discontinuity capturing schemes (DC) to suppress oscillations that may occur in proximity of boundaries or internal layers. Additional improvement in the solution is also obtained when a method that we call FOS (for First-Order Subcells) is used in combination with VMS and DC. In the regions where discontinuities occur, FOS subdivides a spectral element of order p into p ~2 subcells and then uses 1st-order basis functions and integration rules on every subcell of the element. The algorithms are assessed with the solution of classical steady and transient 1D, 2D, and pseudo-3D problems using spectral elements up to order 16.
机译：通过高阶方法准确求解对流占优问题的一个主要问题是求解器保持单调性的能力。这个问题对于频谱元素至关重要，因为吉布斯振荡可能会污染解决方案。但是，与光谱元素一起使用的典型的基于滤波器的稳定技术不是单调的。在本文中，构造了最初为有限元导出的基于残差的稳定方法，并将其应用于高阶谱元素。特别是，我们表明，使用变分多尺度（VMS）方法可以通过减少过冲和下冲而极大地改善输运扩散方程的解，因此可以视为基于滤波器的方案的替代方案。我们还将这些方法与不连续性捕获方案（DC）相结合，以抑制可能在边界或内部层附近发生的振荡。当我们将称为FOS（针对一阶子单元）的方法与VMS和DC结合使用时，解决方案还将获得进一步的改进。在发生不连续的区域中，FOS将阶数为p的频谱元素细分为p〜2个子像元，然后在该元素的每个子像元上使用一阶基函数和积分规则。使用经典的稳态和瞬态1D，2D和伪3D问题的解决方案（使用不超过16级的频谱元素）对算法进行评估。

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