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

a r t i c l e i n f o 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. As it is known, the use of high-order numerical methods for the resolution of compressible flows on unstructured grids is a very complex problem, due to the conflict between keeping the high accuracy and the stabilization the computations. Among all the possible techniques to deal with this kind of flows we can cite Total Variation Diminishing (TVD) methods, essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes. Another different approach is given by shock fitting techniques and multi-resolution methods. are based on the same principle: the addition of extra dissipation to avoid the spurious oscillations near strong gradients. Two of the most popular TVD techniques are flux-limiter methods and slope-limiter methods. Flux-limiter methods [51,61,65] are based on the development of a single numerical flux from two different fluxes. One of them (F l) works well in smooth regions of the flow, and the other one (F h) works well near discontinuities. The single flux has to match with (F l) in smooth regions and with (F h) near discontinuities. On the other hand, slope-limiter methods [4,31,65,67] are based on the limitation of the gradient of the Taylor reconstruction in generalized Godunov's methods. The limitation is obtained by decreasing the value of the gradient near discontinuities or …