Multidimensional WENO-AO Reconstructions Using a Simplified Smoothness Indicator and Applications to Conservation Laws
Multidimensional WENO-AO Reconstructions Using a Simplified Smoothness Indicator and Applications to Conservation Laws
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DOI：
10.1007/s10915-023-02319-x
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发表时间：
2023-08
影响因子：
2.5
通讯作者：
Chieh-Sen Huang;T. Arbogast;Chenyuan Tian
中科院分区：
文献类型：
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作者：
Chieh-Sen Huang;T. Arbogast;Chenyuan Tian
有限体积，加权基本非振荡（WENO）方案需要计算平滑指标。这可能代价高昂，特别是在多个空间维度中。我们考虑使用简单的平滑度指标，即模板中的网格元素数量，是网格元素j上的局部函数平均值，indexm给出目标元素。使用标准WENO加权的重建在此平滑指标下失败。我们开发了一种修正的WENO-Z权重，给出了可靠和准确的自适应顺序重建，我们称之为weno - ao。证明了当解为光滑时，其精度达到大模板多项式近似的阶数，而当解中存在跳变不连续时，其精度降至小模板多项式近似的阶数。在一般网格上一维和二维的数值算例验证了重构的近似性。他们还表明，它在两个空间维度上的重建速度比使用经典平滑指示器的重建速度快10倍左右。将新的重构应用于定义有限体积格式来近似双曲守恒律的解。数值测试结果表明，采用经典平滑度指标的WENO方案与标准WENO方案的计算质量相同，但在二维测试中，计算时间的整体加速约为3.5-5倍。此外，计算效率（CPU时间与错误）显著提高。
Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator, whereis the number of mesh elements in the stencil,is the local function average over mesh elementj, and indexmgives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved.