Machine Learning Changes the Rules for Flux Limiters

Learning to integrate non-linear equations from highly resolved direct numerical simulations (DNSs) has seen recent interest due to the potential to reduce the computational load for numerical fluid simulations. Here, we focus on a specific problem in the integration of fluids: the determination of a flux-limiter for shock capturing methods. Focusing on flux limiters learned from data has the advantage of providing specific plug-and-play components for existing numerical methods. Since their introduction, a large array of flux limiters has been designed. Using the example of the coarse-grained Burgers' equation, we show that flux-limiters may be rank-ordered in terms of how well they integrate various coarse-grainings. We then develop theory to find an optimal flux-limiter and present a set of flux-limiters that outperform others tested for integrating Burgers' equation on lattices with $2\times$, $3\times$, $4\times$, and $8\times$ coarse-grainings. Our machine learned flux limiters have distinctive features that may provide new rules-of-thumb for the development of improved limiters. Additionally, we optimize over hyper-parameters, including coarse-graining, number of discretized bins, and diffusion parameter to find flux limiters that are best on average and more broadly useful for general applications. Our study provides a basis for understanding how flux limiters may be improved upon using machine learning methods.