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- W3194593877 abstract "Learning to integrate non-linear equations from highly resolved direct numerical simulations 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 a theory to find an optimal flux-limiter and present flux-limiters that outperform others tested for integrating Burgers' equation on lattices with 2×, 3×, 4×, and 8× coarse-grainings. We train a continuous piecewise linear limiter by minimizing the mean-squared misfit to six-grid point segments of high-resolution data, averaged over all segments. While flux limiters are generally designed to have an output of ϕ(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." @default.
- W3194593877 created "2021-08-30" @default.
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- W3194593877 date "2022-08-01" @default.
- W3194593877 modified "2023-09-28" @default.
- W3194593877 title "Machine learning changes the rules for flux limiters" @default.
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- W3194593877 doi "https://doi.org/10.1063/5.0102939" @default.
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