Efficient Implementation of Adaptive Order Reconstructions

Abstract

Including polynomials with small degree and stencil when designing very high order reconstructions is surely beneficial for their non oscillatory properties, but may bring loss of accuracy on smooth data unless special care is exerted. 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. CWENOZ-AO -polynomial reconstruction -weighted essentially nonoscillatory -CWENOZadaptive order WENO -finite volume schemes -hyperbolic systemsconservation and balance laws Nq q=0 w q s(R i (t, x q )), where x q and w q are the nodes and weights of a quadrature formula on Ω i .The reconstruction operator should yield an accurate but non oscillatory pointwise approximation of the unknown, based on its cell averages. Beyond the second order of accuracy, one considers essentially non oscillatory reconstructions, which are typically realized by selecting (as in ENO [19]) or more commonly by blending in a nonlinear way polynomials with different degrees and/or stencils. In particular WENO, which was introduced in [33], considers a set of polynomials with equal degree but different stencils and aims at reproducing the accuracy of a higher degree interpolant when the data are locally smooth. The literature on this subject is vast and the reader may refer to [32] for a review.The suboptimal accuracy close to critical points shown by the original design has been later overcome by new definitions of the nonlinear weights (e.g. mapped WENO [20], WENOZ [5,9], the global average weight of [3]) or by taking the small parameter to be dependent on the local mesh size [1].Another source of difficulty in WENO reconstructions is the possible non-existence or nonpositivity of the linear weights for certain grid types and reconstruction points [31]. A quite successful proposal to address this issue was put forward by Levy, Puppo and Russo in [26] for the case of achieving a third order accurate reconstruction at cell center in one and two-dimensional uniform grids. Their CWENO3 reconstruction is a nonlinear blend of one second degree polynomial and of some first degree ones; this approach frees the linear weights from having to satisfy accuracy requirements and allows to choose them arbitrarily, independently of the reconstruction point, independently of the grid type (Ca...

One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

“…) [1,6,23,32,44]. Results about the parameters and the accuracy of , and class reconstructions in various finite volume settings are proven in [13,15,33].…”

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

“…The procedure for the assignment of the linear weights is similar to the CWENO approach described previously. A more sophisticated way to computing the coefficient b, could follow the paradigm of Semplice and Visconti [61] and Cravero et al [95] where a series of comprehensive studies for various parameters, including b and has been performed for obtaining the optimal convergence rates of the designed schemes for structured meshes. Therefore the expansion of this optimisation to unstructured meshes could further fortify the CWENOZ schemes.…”

“…One of the ways to alleviate this is to reduce the size of the directional stencils and consequently the order of approximation from them. This strategy has been adopted by many in developing the sometimes called new, central, compact or cool WENO schemes [1,12,51,53,[60][61][62][63][64][65][66][67][68][69][70]. The key benefit of using these schemes is the reduced computational cost compared to the traditional WENO schemes due to the reduced size of the directional stencils.…”

A new ninth‐order central Hermite weighted essentially nonoscillatory scheme for hyperbolic conservation laws

“…The CWENO construction 12 has been developed by many authors. For instance, in Reference [13], Semplice et al presented a new central WENOZ (CWENOZ) scheme in which the reconstruction polynomial is computed from a single set of nonlinear weights but the linear weights of the polynomials with very low degree. The main advantage of this article is that it gives sufficient conditions on the reconstruction parameter that guarantee the optimal convergence rates on smooth data.…”