Adaptive Mesh Refinement for Hyperbolic Systems Based on Third-Order Compact WENO Reconstruction

Abstract

In this paper we generalise to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy et al. (SIAM J Sci Comput 22(2):656–672, 2000), thus obtaining a truly two-dimensional non-oscillatory third order reconstruction with a very compact stencil and that does not involve mesh-dependent coefficients. This latter characteristic is quite valuable for its use in h-adaptive numerical schemes, since in such schemes the coefficients that depend on the disposition and sizes of the neighbouring cells (and that are present in many existing WENO-like reconstructions) would need to be recomputed after every mesh adaption. In the second part of the paper we propose a third order h-adaptive scheme with the above-mentioned reconstruction, an explicit third order TVD Runge–Kutta scheme and the entropy production error indicator proposed by Puppo and Semplice (Commun Comput Phys 10(5):1132–1160, 2011). After devising some heuristics on the choice of the parameters controlling the mesh adaption, we demonstrate with many numerical tests that the scheme can compute numerical solution whose error decays as $\langle N\rangle ^{-3}$, where $\langle N\rangle$ is the average number of cells used during the computation, even in the presence of shock waves, by making a very effective use of h-adaptivity and the proposed third order reconstruction.