Accuracy of WENO and Adaptive Order WENO Reconstructions for Solving Conservation Laws

Abstract

In this paper, we analyze standard WENO reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO) using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are discontinuities in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions, and compare their performance in solving conservation laws.ARBOGAST, HUANG, AND ZHAO polynomials of different degrees, e.g., as done by Cravero, Puppo, Semplice, and Visconti [5]. The idea was further generalized by Zhu and Qiu [15] and Balsara, Garain, and Shu [2] to define the class of multilevel WENO reconstructions with adaptive order (WENO-AO).In this paper, we analyze the accuracy of the standard WENO and WENO-AO reconstructions. We include results for when the WENO-Z weighting procedure of Castro, Costa, and Don [4, 3] is used. Our results are summarized in Sections 4.3.5 and 5.3 below. The standard WENO reconstruction behaves as desired, even with WENO-Z weights. It is high order accurate when the solution is smooth, and it drops to low order s when there is a discontinuity (not on the central cell), provided only that η ≥ s/2. We show that this condition is sharp. The two-level WENO-AO(r, s) reconstructions behave similarly. They can achieve higher order r accuracy in the smooth case and otherwise maintain at least order s accuracy, provided that when using WENO-JS weights, r ≤ 2s − 1 and η ≥ s/2. WENO-Z weights are more complex, because the accuracy of the reconstruction depends more strongly on the choice of η, as we will show.Multilevel WENO-AO s (r , . . . , r 1 , s) reconstructions [2] are based on a base state with approximation order s. When the solution is smooth the approximation attains the highest order r , but when it is discontinuous, it usually drops to the base level s. When using WENO-JS weights, one requires r ≤ 2s − 1 and η ≥ s/2, which is equivalent to the two-level case. That is, from the point of view of approximation order, there is no point in using the multilevel reconstruction. Again, WENO-Z weights are more complex. There is a restriction on the size of the gap between successive approximation levels, but a careful choice of η can allow any order. However, the accuracy almost always drops to order s when the solution is discontinuous....

Efficient Implementation of Adaptive Order Reconstructions

“…The definition of the CWENOZ-AO reconstruction, see Definition 5, allows to define a very wide set of reconstructions characterized by a very high order gap between the optimal polynomial and lowest degree polynomials. Moreover, we stress the fact that CWENOZ-AO does not require a base level reconstruction, therefore sharing the feature of the WENO-AO reconstruction by [2].…”

“…Before introducing the new adaptive order CWENOZ reconstruction, we recall the definition of the classical CWENO reconstruction operator. Later in this section, CWENOZ and WENO-AO will be introduced, discussing their properties, but for a detailed analysis we point the reader to [11,12,13] for CWENO, to [11,14] for CWENOZ and to [2,4,24] for WENO-AO.…”

“…Adaptive order reconstructions based on the CWENO procedure have been considered in recent years. We focus on the WENO-AO schemes and the formulations given in [4] and [2], recasting them as hierarchic CWENOZ reconstructions. In particular, we focus on the WENO-AO scheme of order 7 as given in [4] since it will be later used for comparisons with our 7-th order adaptive reconstruction introduced in Section 3.2.…”

“…, l. This hierarchic approach effectively enhances the stability of the reconstruction of order r l , but reduces the accuracy suddenly to r 1 whenever a discontinuity is present the stencils of both the polynomials of level l and l − 1. Arbogast, Huang and Zhao in [2] introduced a new hierarchical construction that instead is capable of reducing gradually the accuracy from r l to r 1 as the discontinuity moves inward in the reconstruction stencil. WENO-AO (7,5,3) and WENO-AO (9,7,5,3) are considered in the numerical examples of the papers.…”

“…In this paper we address this issue with a new Central WENOZ (CWENOZ) approach, in which the reconstruction polynomial is computed from a single set of non linear weights, but the linear weights of the polynomials with very low degree (compared to the final desired accuracy) are infinitesimal with respect to the grid size. After proving general results that guide the choice of the CWENOZ parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order WENO reconstructions [4,2]. The novel reconstruction yields similar accuracy and oscillations with respect to the previous ones, but saves up to 20% computational time since it does not rely on a hierarchic approach and thus does not compute multiple sets of nonlinear weights in each cell.Keywords.…”