A Straightforward $hp$-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods (Advances in Applied Mathematics and Mechanics, 2014)
In this paper, high-order Discontinuous Galerkin (DG) method is used to solvethe two-dimensional Euler equations. A shock-capturing method based on the artificialviscosity technique is employed to handle physical discontinuities. Numerical testsshow that the shocks can be captured within one element even on very coarse grids.The thickness of the shocks is dominated by the local mesh size and the local orderof the basis functions. In order to obtain better shock resolution, a straightforward $hp$-adaptivity strategy is introduced, which is based on the high-order contributioncalculated using hierarchical basis. Numerical results indicate that the $hp$-adaptivitymethod is easy to implement and better shock resolution can be obtained with smallerlocal mesh size and higher local order.