Coupled scheme for Hamilton–Jacobi equations

In this paper, we will present some coupled numerical schemes for Hamilton–Jacobi equation by using the scheme proposed in Falcone and Sahu (Coupled scheme for linear and Hamilton-Jacobi-Bellman equations, 2016 [11]). The approach is general and in principle can be applied to couple many different schemes, for example one can couple an accurate method well adepted where the solution is smooth with another method designed to treat discontinuities and/or jumps in the gradients. Clearly, one has to decide where to apply the first or the second method, and this is done by means of a switching parameter which must be computed in every cell at every time step. In this paper, we investigate, in particular, the coupling between an anti-dissipative scheme by Bokanowski and Zidani (J Sci Comput 30(1):1–33 2007, [4]) which has been proposed in order to deal with discontinuous solutions and a semi-Lagrangian scheme by Falcone and Ferretti (Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations. SIAM-Society for Industrial and Applied Mathematics, Philadelphia, 2014 [10]) which is more adept to deal with Lipschitz continuous solutions and is more accurate for regular solutions provided a high-order local interpolation operator is used for the space reconstruction. We will show that how the coupling can be done for two schemes which typically use two different grid reconstructions.