Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields
Efficient WENO-Based Prolongation Strategies for Divergence-Preserving Vector Fields
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DOI：
10.1007/s42967-021-00182-x
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发表时间：
2022-05
影响因子：
1.6
通讯作者：
D. Balsara;S. Samantaray;Sethupathy S.
中科院分区：
文献类型：
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作者：
D. Balsara;S. Samantaray;Sethupathy S.
自适应网格细化（AMR）是在网格层次结构上求解偏微分方程的一种方法，在层次结构的每一级都增加网格细化。AMR层次上的准确治疗需要精确的延长的解决方案从一个粗网格到一个新定义的更精细的网格。对于标量变量，适当的高阶有限体积韦诺方法可以进行这样的扩展。然而，一类偏微分方程，如计算电动力学（CED）和磁流体力学（MHD），要求矢量场保持发散约束。在这种方案中的原始变量由配置在网格的面处的向量场的法向分量组成。因此，对于保持散度约束的向量场的重构和延拓策略必然更加复杂。在本文中，我们提出了一个四阶发散约束保持延长策略，是解析精确的。使用解析精确方法扩展到高阶是非常具有挑战性的。为了克服这一挑战，发明了一种新的类似WENO的重建策略，该策略匹配面中矢量场的矩，其中矢量场分量是同位的。这种方法几乎是保持发散约束的，因此我们称之为WENO-ADP。为了使它完全发散约束保持，一个触摸的程序开发的基础上，约束最小二乘法（CLSQ）的方法恢复发散约束的机器精度。经过修饰后，它被称为WENO-ADPT。结果表明，细化比的两个和更高的可以容纳。在这项工作中，一个更广泛的兴趣是，我们也已经能够发明非常efficientfinite volumeWENO方法，其中的系数很容易获得和多维平滑指标可以表示为完美的平方。我们证明了发散约束保持策略的作品在几个高订单的发散自由矢量场以及矢量场，矢量场的发散必须匹配的电荷密度和它的高阶矩。我们还表明，我们的方法克服了晚时间的不稳定性，已被称为困扰自适应计算CED。
Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly defined finer mesh. For scalar variables, suitably high-order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, such as computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this paper we present a fourth-order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces, where the vector field components are collocated. This approach is almost divergence constraint-preserving, therefore, we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficientfinite volumeWENO methods, where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields, where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in CED.