论文标题
卷曲约束保留重建及其为模拟方案设计提供的指导
Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design
论文作者
论文摘要
已知几种重要的PDE系统，例如磁水动力学和计算电动力学，可以支持矢量场的差异在无差异或发散约束提供的方式中演变。 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.由于用于以前类PDE的溶液的模拟数值方案已经发达了，因此我们从它们中汲取了指导，以解决后一种PDES的解决方案。我们表明，对卷曲约束的重建的研究使我们对这些涉及PDE的一致，模拟方案的设计的设计有很大的见解。还记录了多维Riemann求解器在促进此类方案设计方面的重要性。我们研究了与模拟不连续的盖尔金（DG）和有限体积（FV）方案的设计有关的PDE的设计，因此研究了卷曲约束的重建问题。这是针对两个和三维结构化的网格问题完成的，我们为重建提供了封闭形式的表达式。还讨论了这种重建在自适应网状细化（AMR）中无卷曲或弯曲的载体延长（AMR）中的作用。在两个维度中，还提出了对结构保存的DG样方案的von Neumann分析，该方案在模仿卷曲约束上。还提出了数值结果，以表明这些方案符合其设计准确性。
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.
