An efficient class of WENO schemes with adaptive order for unstructured meshes
An efficient class of WENO schemes with adaptive order for unstructured meshes
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DOI：
10.1016/j.jcp.2019.109062
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发表时间：
2020-03
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影响因子：
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通讯作者：
D. Balsara;S. Garain;V. Florinski;W. Boscheri
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文献类型：
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作者：
D. Balsara;S. Garain;V. Florinski;W. Boscheri
双曲守恒定律的有限差分 WENO 方案的最新进展导致了具有自适应精度顺序的 WENO 方案。例如，当五阶模板中的解的平滑性保证时，WENO-AO(5,3) 方案可以提供高达五阶的精度，然而，当网格上的解不能保证更高阶时，它可以自适应地下降到三阶精度。拥有双曲守恒定律的有限体积 WENO 方案的类似功能，尤其是在非结构化网格上，可能非常有价值。本文记录了非结构化网格的有限体积 WENO-AO(4,3) 和 WENO-AO(5,3) 方案的设计。与结构化网格的 WENO-AO 一样，关键的进步在于认识到存在一个有利的基础集，该基础集非常容易构建，并且计算大大简化。与有限差分 WENO 一样，我们意识到可以在大型、居中、非常高精度的模板和低阶中心 WENO 方案之间进行非线性混合，但该方案非常稳定并且能够捕获物理上有意义的极值。这产生了一类在非结构化网格上运行良好的自适应阶 WENO 方案。在大型和小型模板上，由于选择了有利的泰勒级数基础，我们能够使模板评估步骤非常有效。通过扩展平行轴定理，我们表明有限体积重建得到了显着的简化。我们的方法不需要解决受约束的最小二乘问题，而是只需要在每个模板上解决较小的最小二乘问题。这也简化了每个模板的矩阵组装和解决方案。平滑度指标的评估也得到简化。精度测试表明该方法满足设计精度。提出了几个严格的测试问题来证明该方法非常稳健且运行良好。选择测试问题是为了表明我们的方法可以应用于许多不同的网格，这些网格用于映射几何复杂性或解决方案复杂性。
Recent advances in finite-difference WENO schemes for hyperbolic conservation laws have resulted in WENO schemes with adaptive order of accuracy. For instance, a WENO-AO(5,3) scheme can provide up to fifth order of accuracy when the smoothness of the solution in the fifth order stencil warrants it, and yet, it can adaptively drop down to third order of accuracy when the higher order is not warranted by the solution on the mesh. Having an analogous capability for finite-volume WENO schemes for hyperbolic conservation laws, especially on unstructured meshes, can be very valuable. The present paper documents the design of finite volume WENO-AO(4,3) and WENO-AO(5,3) schemes for unstructured meshes. As with WENO-AO for structured meshes, the key advance lies in realizing that there is a favorable basis set, which is very easily constructed, and in which the computation is dramatically simplified. As with finite-difference WENO, we realize that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order central WENO scheme that is, nevertheless, very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes that work well on unstructured meshes. On both the large and small stencils we have been able to make the stencil evaluation step very efficient owing to the choice of a favorable Taylor series basis. By extending the Parallel Axis Theorem, we show that there is a significant simplification in the finite volume reconstruction. Instead of solving a constrained least squares problem, our method only requires the solution of a smaller least squares problem on each stencil. This also simplifies the matrix assembly and solution for each stencil. The evaluation of smoothness indicators is also simplified. Accuracy tests show that the method meets its design accuracy. Several stringent test problems are presented to demonstrate that the method works very robustly and very well. The test problems are chosen to show that our method can be applied to many different meshes that are used to map geometric complexity or solution complexity.