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

TL;DR

This paper introduces a new third-order non-oscillatory reconstruction method for hyperbolic systems on adaptive non-uniform grids, enabling efficient and accurate solutions even with shocks.

Contribution

It generalizes Compact WENO to non-uniform quad-tree grids and develops an adaptive scheme with proven error decay, improving computational efficiency and accuracy.

Findings

Error decays as $ extless N extgreater^{-3}$ with mesh adaptation

Effective shock capturing with third-order accuracy

Reconstruction avoids mesh-dependent coefficients

Abstract

In this paper we generalize to non-uniform grids of quad-tree type the Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput., 2001), 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 neighboring 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., 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.