Enhanced fifth order WENO shock-capturing schemes with deep learning

Abstract

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method. Moreover, the formal order of accuracy of the resulting scheme can be proven.We demonstrate our findings with the inviscid Burgers’ equation, the Buckley–Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks. Finally, the straight-forward extension of the method to two-dimensional problems is included and illustrated using the 2D Burgers’ equation.