Stabilized P²-DG method with artificial viscosity for steady hyperbolic conservation laws

It is well-known that convergence to steady state is hard when a high order discontinuous Galerkin (DG) method is applied for transonic and supersonic flows in polynomial space even with post-processing such as limiters and positivity preservation. The method discretizes the hyperbolic conservation laws in space in advance to obtain a system of first-order ordinary differential equations in time. As a result, steady-state solution of the DG method is equivalent to the equilibrium point of this system. In this paper, we analyze the stability of DG methods in the view point of dynamical systems for the scalar conservation law. We show that the steady-state solution of the 3rd-order DG method is not always stable in the presence of shock waves, and then we propose an artificial viscosity to stabilize the DG method and shows that the artificial viscosity has to be order one of the mesh size to improve stability. To maintain higher order accuracy, the proposed artificial viscosity is only applied in the vicinities of shock waves together with a shock-wave indicator. Numerical results are given to verify theoretical analysis. Several transonic/supersonic flow test cases are also present to demonstrate the effectiveness of the present artificial viscosity.