A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations
A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations
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DOI：
10.1016/j.jcp.2017.11.035
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发表时间：
2016-12
期刊：
影响因子：
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通讯作者：
Marian Piatkowski;Steffen Müthing;P. Bastian
中科院分区：
文献类型：
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作者：
Marian Piatkowski;Steffen Müthing;P. Bastian
本文在投影方法的框架下研究不可压Navier-Stokes方程的间断Galerkin（DG）方法。特别是，我们采用对称内部惩罚DG方法内的二阶旋转增量压力校正计划。本文的主要重点是三个方面：i）我们提出了一个修改的迎风格式的基础上Vijayasundaram数值通量，具有良好的性能，在DG的上下文中。ii）我们在亥姆霍兹投影步骤中提出了一种新的后处理技术，该技术基于局部计算的压力校正的H（div）重建，是离散设置中的投影，并确保投影速度精确地满足离散连续性方程。因此，它还提供了投影速度的局部质量守恒。iii）数值结果表明了该格式在不同多项式次数下的性质，它不仅适用于已知解的二维问题，也适用于大规模的三维问题。特别是，我们解决二阶收敛时间的分裂计划，以及它的长期稳定性。
In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier–Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on H (div) reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.