Characteristic modal shock detection for discontinuous finite element methods

Abstract

Within the context of high-order discontinuous finite element methods and their applications to high-speed flows with shocks and discontinuities, modifications involving the use of characteristic variables are discussed in order to improve the behaviour of the shock-capturing approach in the particular case that a modal shock-sensor is adopted. The proposed technique—surprisingly simple and yet, to the author’s knowledge, previously unexplored—achieves enhanced shock detection, leading to smoother distributions of the added shock-capturing viscosity across the shock fronts, even in the multidimensional case and without any noticeable degradation in terms of its localisation and strength. Numerical results show that using characteristic variables for shock detection in the formulation of the modal shock-sensor is particularly efficient in mitigating, or even suppressing, post-shock oscillations in the case of extremely strong shocks undergoing mutual interactions or colliding with solid boundaries. At the same time, characteristic based sensors appear to be particularly robust and well suited to alleviate the need of a precise fine tuning of the shock-capturing parameters.