A reconstructed discontinuous Galerkin method based on a Hierarchical WENO reconstruction for compressible flows on tetrahedral grids

A reconstructed discontinuous Galerkin (RDG) method based on a hierarchical WENO reconstruction, termed HWENO (P1P2) in this paper, designed not only to enhance the accuracy of discontinuous Galerkin methods but also to ensure the nonlinear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this HWENO (P1P2) method, a quadratic polynomial solution (P-2) is first reconstructed using a Hermite WENO reconstruction from the underlying linear polynomial (P-1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the RDG method. The developed HWENO (P1P2) method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, robustness, and non-oscillatory property. The numerical experiments indicate that the HWENO (P1P2) method is able to capture shock waves within one cell without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method

Detecting troubled-cells on two-dimensional unstructured grids using a neural network

In a recent paper [Ray and Hesthaven, J. Comput. Phys. 367 (2018), pp 166-191], we proposed a new type of troubled-cell indicator to detect discontinuities in the numerical solutions of one-dimensional conservation laws. This was achieved by suitably training an articial neural network on canonical local solution structures for conservation laws. The proposed indicator was independent of problem-dependent parameters, giving it an advantage over existing limiter-based indicators. In the present paper, we extend this approach to train a similar network capable of detecting troubled-cells on two-dimensional unstructured grids. The proposed network has a smaller architecture compared to its one-dimensional predecessor, making it computationally efficient. Several numerical results are presented to demonstrate the performance of the new indicator