In this paper, high-order Discontinuous Galerkin (DG) method is used to solve
the two-dimensional Euler equations. A shock-capturing method based on the artificial
viscosity technique is employed to handle physical discontinuities. Numerical tests
show 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 order
of the basis functions. In order to obtain better shock resolution, a straightforward
hp-adaptivity strategy is introduced, which is based on the high-order contribution
calculated using hierarchical basis. Numerical results indicate that the hp-adaptivity
method is easy to implement and better shock resolution can be obtained with smaller
local mesh size and higher local order.