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

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 RungeKutta 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 N −3 , where N 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.

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

“…Next, we will describe first the high order discretization L i (u(t)) of the spatial operator on the right hand side of (5) and then the time discretization will be achieved with a third order time accurate Runge-Kutta (RK3) scheme maintaining, at the same 4 time, better than second order of accuracy in space and time. At a difference from [SCR15,CS15], here stabilization of the high accurate reconstructions is obtained by means of an a posteriori MOOD limiting [CDL11a,DCL12,DLC13,LDD14] under the classical CFL condition of a RK3 scheme. At last an adaptive mesh refinement (AMR) technique is employed [SCR15] to enhance even further the accuracy of the overall scheme.…”

“…At a difference from [SCR15,CS15], here stabilization of the high accurate reconstructions is obtained by means of an a posteriori MOOD limiting [CDL11a,DCL12,DLC13,LDD14] under the classical CFL condition of a RK3 scheme. At last an adaptive mesh refinement (AMR) technique is employed [SCR15] to enhance even further the accuracy of the overall scheme. Next sub-sections describe the main aspects of this numerical method.…”

“…Following [SCR15], we have introduced the constant terms in Ψ i,3 and Ψ i,4 so that all basis functions have null cell average. The reconstruction on element Ω i requires a so-called reconstruction stencil S i , that is an appropriate set including Ω i and a number of its neighbors that we denote as…”

“…This technique is known under the acronym AMR and many improvements and publications are available since the 80's [BO84,BC89]. A drastic reduction of the costs without sacrificing the level of accuracy is the main reason why those techniques were developed and are still in use today [SCR15]. In this work we rely on AMR technology using the so-called numerical entropy production refinement criterion [Pup04,PS11,SCR15], with a subtle but important difference compared to classical AMR procedures: each timestep is performed with the best possible refined mesh driven by the entropy production between t n and t n+1 .…”

“…A drastic reduction of the costs without sacrificing the level of accuracy is the main reason why those techniques were developed and are still in use today [SCR15]. In this work we rely on AMR technology using the so-called numerical entropy production refinement criterion [Pup04,PS11,SCR15], with a subtle but important difference compared to classical AMR procedures: each timestep is performed with the best possible refined mesh driven by the entropy production between t n and t n+1 . In other words, if the numerical solution associated to its AMR mesh at time t n+1 could be improved (dixit the entropy production criterion) then, the mesh is appropriately refined, and, second, the solution is sent back at t n for re-computation.…”