We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework.
翻译:我们建议使用一个新的结构保存不连续 Galerkin (SPDG) 操作器, 该操作器在离散级别上恢复与矢量字段曲线差异有关的代数属性, 通常被称为 div- curl 问题 。 3D 采用错开的 Cartesian 网格, 矢量字段在控制量的角处自然定义, 而其卷轴被评价为以单元格为中心的数量 。 首先, 卷轴操作器被重写成一个调色器的差异, 从而可以设计兼容的有限差异方案, 并被证明可以模拟向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向, 向向向向向向向向向向向向向向向向向向向向向向向向向向向向向,向向向向向向向向向向向,向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向方向轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉轉的