Learning to integrate non-linear equations from highly resolved direct numerical simulations (DNSs) has seen recent interest for reducing the computational load for fluid simulations. Here, we focus on determining a flux-limiter for shock capturing methods. Focusing on flux limiters provides a specific plug-and-play component for existing numerical methods. Since their introduction, an array of flux limiters has been designed. Using the coarse-grained Burgers' equation, we show that flux-limiters may be rank-ordered in terms of their log-error relative to high-resolution data. We then develop theory to find an optimal flux-limiter and present flux-limiters that outperform others tested for integrating Burgers' equation on lattices with $2\times$, $3\times$, $4\times$, and $8\times$ coarse-grainings. We train a continuous piecewise linear limiter by minimizing the mean-squared misfit to 6-grid point segments of high-resolution data, averaged over all segments. While flux limiters are generally designed to have an output of $\phi(r) = 1$ at a flux ratio of $r = 1$, our limiters are not bound by this rule, and yet produce a smaller error than standard limiters. We find that our machine learned limiters have distinctive features that may provide new rules-of-thumb for the development of improved limiters. Additionally, we use our theory to learn flux-limiters that outperform standard limiters across a range of values (as opposed to at a specific fixed value) of coarse-graining, number of discretized bins, and diffusion parameter. This demonstrates the ability to produce flux limiters that should be more broadly useful than standard limiters for general applications.
翻译:摘要:学习如何从高度分辨率的直接数值模拟中积分非线性方程已经引起了人们的极大兴趣,因为这可以减少流体模拟的计算负荷。在这里,我们专注于为激波捕获方法确定流量限制器。专注于流量限制器为现有的数值方法提供了一个特定的即插即用组件。自从它们被引入以来,各种各样的流量限制器已经被设计出来。使用粗粒化的Burgers方程,我们展示流量限制器可以按照它们相对于高分辨率数据的对数误差进行排名。然后,我们开发了理论以找到最佳的流量限制器,并呈现了在$2\times$、$3\times$、$4\times$和$8\times$粗粒化的晶格上集成Burgers方程的流量限制器,这些限制器优于经过测试的其他限制器。我们通过最小化高分辨率数据的6个网格点段的均方误差来训练一个连续分段线性限制器,并将其应用于流量上限,从而学习一个连续分段线性限制器。虽然流量限制器通常设计为在流量比$r=1$时输出$\phi(r)=1$,但我们的限制器不受这个规则的束缚,但却产生比标准限制器更小的误差。我们发现,我们的机器学习限制器具有独特的特点,可以为发展改进的限制器提供新的经验规则。此外,我们使用我们的理论来学习流量限制器,在粗细程度、离散化的箱数以及扩散参数的一系列数值上优于标准限制器。这表明,我们可以产生比标准限制器更广泛适用于一般应用的流量限制器。