Abstract
Recent works have shown that neural networks are promising parameter-free limiters for a variety of numerical schemes (Morgan et al. in A machine learning approach for detecting shocks with high-order hydrodynamic methods. https://doi.org/10.2514/6.2020-2024; Ray et al. in J Comput Phys 367: 166–191. https://doi.org/10.1016/j.jcp.2018.04.029, 2018; Veiga et al. in European Conference on Computational Mechanics and VII European Conference on Computational Fluid Dynamics, vol. 1, pp. 2525–2550. ECCM. https://doi.org/10.5167/uzh-168538, 2018). Following this trend, we train a neural network to serve as a shock-indicator function using simulation data from a Runge-Kutta discontinuous Galerkin (RKDG) method and a modal high-order limiter (Krivodonova in J Comput Phys 226: 879–896. https://doi.org/10.1016/j.jcp.2007.05.011, 2007). With this methodology, we obtain one- and two-dimensional black-box shock-indicators which are then coupled to a standard limiter. Furthermore, we describe a strategy to transfer the shock-indicator to a residual distribution (RD) scheme without the need for a full training cycle and large dataset, by finding a mapping between the solution feature spaces from an RD scheme to an RKDG scheme, both in one- and two-dimensional problems, and on Cartesian and unstructured meshes. We report on the quality of the numerical solutions when using the neural network shock-indicator coupled to a limiter, comparing its performance to traditional limiters, for both RKDG and RD schemes.
Abgrall, Remi