Mitigation of post-shock oscillations induced by artificial viscosity in discontinuous finite element methods

Current shock-capturing techniques for high-order discontinuous finite element methods based on modal energy shock sensors coupled to an artificial viscosity operator provide efficient and relatively simple solutions to tackle high-speed flows with discontinuities. Yet, due to inherent inhomogeneities in modal detection of shocks within elements, post-shock spurious oscillations can occur. These can be particularly severe in the case of slow-moving shocks. In the present paper, the origin of such oscillations is identified in the way the artificial viscosity is distributed across shocks by the modal shock detector. A novel approach is hence proposed to mitigate post-shock oscillations by introducing, via exponential averaging, a time delay in the artificial viscosity injection. Numerical one- and two-dimensional tests will show that a calibration of the time delay based on the shock propagation velocity can suppress, almost completely, post-shock oscillations, even in the most severe cases. Inspired by machine learning techniques, an efficient way to evaluate the propagation velocities of different shocks is also proposed, leading to a method to suppress post-shock oscillations, induced by the artificial viscosity, which is extremely simple, and relatively inexpensive.