A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations

A shock-capturing methodology is developed for non-linear computations using low-dissipation schemes and centered finite differences. It consists in applying an adaptative second-order filtering to handle discontinuities in combination with a background selective filtering to remove grid-to-grid oscillations. The shock-capturing filtering is written in its conservative form, and its magnitude is determined dynamically from the flow solutions. A shock-detection procedure based on a Jameson-like shock sensor is derived so as to apply the shock-capturing filtering only around shocks. A second-order filter with reduced errors in the Fourier space with respect to the standard second-order filter is also designed. Linear and non-linear 1D and 2D problems are solved to show that the methodology is capable of capturing shocks without providing dissipation outside shocks. The shock detection allows in particular to distinguish shocks from linear waves, and from vortices when it is performed from dilatation rather than from pressure. Finally the methodology is simple to implement and reasonable in terms of computational cost.