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首页> 外文期刊>Applied numerical mathematics >Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids
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Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier-Stokes equations on staggered Cartesian grids

机译：交错笛卡尔网格上不可压缩的Navier-Stokes方程的谱半隐式和时空不连续Galerkin方法

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摘要

In this paper two new families of arbitrary high order accurate spectral discontinuous Galerkin (DG) finite element methods are derived on staggered Cartesian grids for the solution of the incompressible Navier-Stokes (NS) equations in two and three space dimensions. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. Thanks to the use of a nodal basis on a tensor-product domain, all discrete operators can be written efficiently as a combination of simple one-dimensional operators in a dimension-by-dimension fashion. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor θ ∈ [0.5,1] for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. From our numerical experiments we find that the pressure system appears to be reasonably well-conditioned, since in all test cases shown in this paper the use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the Navier-Stokes equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly using again a staggered mesh approach, where the viscous stress tensor is also defined on the dual mesh.The second family of staggered DG schemes proposed in this paper achieves high order of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polynomials. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterative Picard procedure is introduced, which leads to a space-time pressure-correction algorithm. In this manner, the symmetry and positive definiteness of the pressure system are not compromised. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. These features are typically not easy to obtain all at the same time for a numerical method applied to the incompressible Navier-Stokes equations. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N = 11, using a large set of non-trivial test problems in two and three space dimensions, for which erither analyticl, numerical or experimental reference solutions exist.
机译：本文在交错的笛卡尔网格上推导了两个新的任意高阶精确谱不连续伽勒金（DG）有限元方法，用于求解二维和三维空间中不可压缩的Navier-Stokes（NS）方程。压力和速度的离散解以沿着不同网格的分段多项式的形式表示。在主网格的控制体积上定义压力的同时，在基于边缘的双重控制体积上定义速度分量，从而形成空间交错的网格。由于在张量积域上使用了节点基础，因此所有离散算子都可以按逐维方式作为简单的一维算子的组合有效地写入。在第一个族中，仅在空间中才能获得较高的精度，而通过在动量方程中为压力梯度引入隐式因子θ∈[0.5,1]，可以得出简单的半隐式时间离散。将离散动量方程式替换为连续性方程式的弱形式后，即可实现交错的真正优势。实际上，所得到的压力线性系统是对称且为正定的，并且可以是五角形（在2D中）或七角形（在3D中）。结果，可以通过经典的无矩阵共轭梯度法非常有效地求解压力系统。从我们的数值实验中，我们发现压力系统似乎条件良好，因为在本文所示的所有测试案例中，都不需要使用预处理器。在Navier-Stokes方程的现有隐式DG方案中，这是一个相当独特的功能。为了避免由于粘性项引起的稳定性限制，再次使用交错网格方法隐式离散后者，其中在双重网格上也定义了粘性应力张量。本文提出的第二类交错DG方案实现了通过使用分段时空多项式表示数值解，还可以在时间上实现较高的精度。为了避免所采用的分数步进的低精度，引入了一种简单的迭代Picard程序，这导致了时空压力校正算法。以这种方式，压力系统的对称性和正确定性不受影响。所得算法稳定，计算效率高，并且同时在空间和时间上都具有任意高阶精度。对于应用于不可压缩的Navier-Stokes方程的数值方法，通常很难同时获得所有这些特征。新的数值方法已经针对高达N = 11的近似多项式进行了充分的验证，使用了在两个和三个空间维度上的大量非平凡测试问题，对于这些问题，还存在解析，数值或实验参考解决方案。

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