On the Accuracy of WENO and CWENO Reconstructions of Third Order on Nonuniform Meshes

Abstract

Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called , that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing as a function of the local mesh size hj. In particular we show that choosing = h 2 j or = hj is beneficial for the error and convergence order, studying on several nonuniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.

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

“…Moreover the generated oscillating solution would spoil the refining and coarsening procedures of the AMR framework. In previous works, CWENO polynomial reconstructions have been used to limit and stabilize this scheme [PS11,SCR15,CS15]. In this work we will use an a posteriori stabilization procedure based on a troubled cell detector and subsequent re-computations with a second-or first-order finite volume scheme.…”

“…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.…”

Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction

“…The presence of a positive ϵ at the denominator in the previous formula is essential in order to avoid a division by zero in the case of reconstruction of very flat data. However, in the one‐dimensional case it was proven (for both WENO and CWENO) that the value of ϵ can affect the convergence rate close to local extrema. Intuitively, when a local extrema is present in the stencil, some of the indicators will be of size and some others will be ≈ h 2 : unless ϵ is at least as big as h 2 , formula will bias the nonlinear weights toward the lower‐order polynomials with smaller indicators, effectively leading to an order reduction.…”

“…After the seminal paper, the one‐dimensional CWENO technique was extended to fifth order, the properties of the third‐order versions were studied in detail on uniform meshes and nonuniform ones, and finally, arbitrary high‐order variants were introduced …”

“…After the seminal paper, 11 the one-dimensional CWENO technique was extended to fifth order, 13,14 the properties of the third-order versions were studied in detail on uniform meshes 15 and nonuniform ones, 16 and finally, arbitrary high-order variants were introduced. 17,18 The CWENO procedure in more than one space dimension has been first employed in the form of a combination of a central parabola in two variables and four linear polynomials.…”