RKDG方案中基于子小区的故障区指示器和一类新的混合RKDG + HWENO方案
A Sub-Cell Based Indicator for Troubled Zones in RKDG Schemes and a Novel Class of Hybrid RKDG+HWENO Schemes
摘要：
Runge Kutta间断Galerkin（RKDG）格式可以为一类重要的科学问题提供高精度的解。用它们来解决冲击和其他不连续性问题，需要人们有一个检测这些不连续性存在的策略。为了克服以往限制器设计的局限性，我们认识到RKDG格式的解可以在一个区域内携带有意义的子结构，而这个区域可能不需要限制。为了使这个子结构可见，我们采用子单元的方法来检测具有不连续性的区域，称为问题区域。单调性保持（MP）策略被用来区分有意义的子结构和冲击。该策略不会不加选择地修剪极值，但是，无标度和问题无关。使用加权基本无振荡（WENO）或Hermite WENO（HWENO）方法校正故障区域的矩。在做这项工作的过程中，它也意识到，在解决方案中最显着的变化是包含在解决方案的变量和他们的第一时刻。因此，它表明，可以使用的变量和它们的第一时刻，在一个非常可观的节省计算机内存和没有任何损失的准确性顺序重建的附加时刻。我们称这种方案为混合RKDG+HWENO方案。明确的，易于实施的公式已被编目的第三和第四阶的空间精度。还提出了几个严格的测试问题。
英文摘要：
Runge Kutta Discontinuous Galerkin (RKDG) schemes can provide highly accurate solutions for a large class of important scientific problems. Using them for problems with shocks and other discontinuities requires that one has a strategy for detecting the presence of these discontinuities. In order to get past previous limitations in designing a limiter, we realize that the solution in RKDG schemes can carry meaningful sub-structure within a zone that may not need to be limited. To make this sub-structure visible, we take a sub-cell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful sub-structure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scale-free and problem-independent. The moments of the troubled zones is corrected using a weighted essentially non-oscillatory (WENO) or Hermite WENO (HWENO) approach. In the course of doing this work it was also realized that the most significant variation in the solution is contained in the solution variables and their first moments. Thus it is shown that the additional moments can be reconstructed using the variables and their first moments, resulting in a very substantial savings in computer memory and without any loss in order of accuracy. We call such schemes hybrid RKDG+HWENO schemes. Explicit, easy to implement formulae have been catalogued for third and fourth order of spatial accuracy. Several stringent test problems have also been presented.
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