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:学习来自高分辨率直接数字模拟( DNSs) 的非线性方程式, 从高清晰度直接数字模拟( DNSs) 学习整合非线性方程式, 近来人们对减少液体模拟的计算负荷感兴趣。 在这里, 我们侧重于确定冲击捕捉方法的通量限制值。 聚焦通量限制器为现有数字方法提供了特定的插座和游戏部分。 自引入以来, 设计了一系列通量限制器。 使用粗糙的汉堡方程式, 我们显示, 通量限制器可能是按其对高清晰度数据的日志- 错误值排序。 我们随后开发了理论, 以找到一个最佳通量限制器, 并展示出一个超越其他测试的通量限制值的通量限制值。 通量限制器一般设计为2美元, 3美元, 4美元, 8美元, 粗略度限制值。 普通限值, 我们训练一个连续的线性线性限值限制器, 以最小值为六里格点, 普通解限值为6, 普通的限值, 而我们的标准值的流限值为1美元, 我们的流限值的流限值的递值比标准值的递增值, 标准值为1美元。</s>