Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of ``adaptive filtered'' schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function $F$ and by a switching parameter $\varepsilon^n=\varepsilon^n({\Delta t,\Delta x})>0$ which goes to 0 as $(\Delta t,\Delta x)$ is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter $\varepsilon^n_j$ in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme.