Simulating shocks and discontinuities by solving the spatially-filtered Euler equations

To ensure numerical stability in the vicinity of shocks, a variety of methods have been used. These methods include shock capturing schemes such as weighted essentially non-oscillatory (WENO) schemes, as well as the addition of artificial diffusivities to the governing equations. Centered finite difference schemes are often avoided near discontinuities due to the tendency for significant oscillations. However, such schemes have desirable conservation properties compared to many shock-capturing schemes. The objective of this work is to perform accurate and stable simulations of discontinuities by deriving diffusion terms from first principle and then applying these analytical terms within a centered differencing framework. The physical Euler equations are filtered, and a sub-filter scale term for the momentum equation is extracted specifically for a shock. No sub-filter scale terms are required for the continuity, energy, or species equations. Rather, there is a minimum amount of numerical diffusion required for the energy and species equations. This approach is tested for a variety of problems involving shocks and discontinuities.