Curl约束保持重构及其对仿生方案设计的指导作用
Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design
摘要：
几个重要的偏微分方程系统，如磁流体力学和计算电动力学，已知支持对合，其中向量场的发散以无发散或发散约束保持的方式演化。最近，出现了新的一类PDE系统的超弹性，可压缩多相流，所谓的爱因斯坦场方程的一阶约化，或一种新的一阶双曲薛定谔方程的重新制定，仅举几例，其中PDE中的对合支持无旋或旋度约束保持向量场的演化。由于前一类偏微分方程的解决方案的模拟数值计划是发达的，我们从中获得指导后一类偏微分方程的解决方案。我们表明，旋度约束保持重建的研究给我们提供了大量的洞察力，这些对合偏微分方程的一致，模仿计划的设计。多维黎曼解算子的重要性，在方便设计这样的计划也记录。我们研究了旋度约束保持重建的问题，因为它涉及到设计的模拟不连续Galerkin（DG）和有限体积（FV）计划的偏微分方程，支持这样的对合。这是做二维和三维结构网格问题，我们提供封闭形式的重建表达式。本文还讨论了这种重建在自适应网格细化（AMR）中向量场的无卷曲或保持卷曲的延伸中所起的作用。在二维空间中，对满足旋度约束的保结构DG格式进行了冯·诺依曼分析。数值结果也表明，该格式满足其设计精度。
英文摘要：
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first order reductions of the Einstein field equations, or a novel first order hyperbolic reformulation of Schrödinger's equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. Since mimetic numerical schemes for the solution of the former class of PDEs are well-developed, we draw guidance from them for the solution of the latter class of PDEs. We show that a study of the curl constraint-preserving reconstruction gives us a great deal of insight into the design of consistent, mimetic schemes for these involutionary PDEs. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic discontinuous Galerkin (DG) and finite volume (FV) schemes for PDEs that support such an involution. This is done for two and three dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The role that this reconstruction plays in the curl-free, or curl-preserving prolongation of vector fields in adaptive mesh refinement (AMR) is also discussed. In two dimensions, a von Neumann analysis of structure-preserving DG-like schemes that mimetically satisfy the curl constraints, is also presented. Numerical results are also presented to show that the schemes meet their design accuracy.
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