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 \(\mathsf {WENOZ}\) (\(\mathsf {CWENOZ}\)) 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 \(\mathsf {CWENOZ}\) parameters, we study a concrete example of a reconstruction that blends polynomials of degree six, four and two, mimicking already published Adaptive Order \(\mathsf {WENO}\) reconstructions (Arbogast et al. in SIAM J Numer Anal 56(3):1818-1947, 2018),(Balsara et al. in J Comput Phys 326:780-804, 2016). 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.

中文翻译：

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自适应顺序重构的有效实现

摘要

在设计非常高阶的重构时，包括小阶数和模版的多项式肯定会因其非振荡特性而受益，但除非特别注意，否则可能会导致平滑数据的准确性下降。在本文中，我们使用新的Central \（\ mathsf {WENOZ} \）（\（\ mathsf {CWENOZ} \））方法来解决此问题，该方法是根据一组非线性权重来计算重构多项式，但是相对于网格大小，非常低的多项式的线性权重（与最终所需的精度相比）是极小的。在证明指导选择\（\ mathsf {CWENOZ} \）的一般结果之后参数，我们研究了一个混合了六阶，四阶和二阶多项式的重构的具体示例，模仿了已发布的Adaptive Order \（\ mathsf {WENO} \）重构（Arbogast等人，在SIAM J Numer Anal 56（3）中） ：1818-1947，2018），（Balsara等人，J Comput Phys 326：780-804，2016）。相对于以前的方法，这种新颖的方法可产生相似的精度和振荡，但由于它不依赖于分层方法，因此可节省多达20％的计算时间，因此不会在每个像元中计算多组非线性权重。