Machine Learning Changes the Rules for Flux Limiters

Abstract

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 $\varphi (r)=1$ 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.

arXiv commentcleaned up erratum: one corrected figure and some minor text updates

References in corpus (5)

• Enhancing Computational Fluid Dynamics with Machine Learning
• Learning data driven discretizations for partial differential equations
• Accelerating Eulerian Fluid Simulation With Convolutional Networks
• Compressed Convolutional LSTM: An Efficient Deep Learning framework to Model High Fidelity 3D Turbulence
• Leveraging Bayesian Analysis To Improve Accuracy of Approximate Models

Cited by in corpus (1)

• Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws