A new shock-capturing technique based on Moving Least Squares for higher-order numerical schemes on unstructured grids

Abstract

This paper presents a shock detection technique based on Moving Least Squares reproducing kernel approximations. The multiresolution properties of these kinds of approximations allow us to define a wavelet function to act as a smoothness indicator. This MLS sensor is used to detect the shock waves. When the MLS sensor is used in a finite volume framework in combination with slope limiters, it improves the results obtained with the single application of a slope-limiter algorithm. The slope-limiter algorithm is activated only at points where the MLS sensor detects a shock. This procedure results in a decrease of the artificial dissipation introduced by the whole numerical scheme. Thus, this new MLS sensor extends the application of slope limiters to higher-order methods. Moreover, as Moving Least Squares approximations can handle scattered data accurately, the use of the proposed methodology on unstructured grids is straightforward. The results are very promising, and comparable to those of essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes. Another advantage of the proposed methodology is its multidimensional character, that results in a very accurate detection of the shock position in multidimensional flows.