Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure

Liyang Xu,Yongquan Feng,Xiaoguang Ren,Xinhai Xu,Yunrui Guo,Xuejun Yang

doi:10.1177/1687814019855586

Liyang Xu, Yongquan Feng + Show 4 more

Open Access

https://doi.org/10.1177/1687814019855586

Copy DOI

Abstract

In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.

Highlights

Discontinuous Galerkin method (DGM) is one of the most potential high-order discretization method among the state-of-the-art methods, such as finite difference methods and finite volume method (FVM)
We find that sufficient fine mesh and high polynomial order help to make discontinuous Galerkin (DG)-continuous Galerkin (CG) scheme more robust and accurate for high Reynolds number flow
We analyzed in detail the stability of the splitting method with discontinuous velocity and continuous pressure

Summary

Introduction

Discontinuous Galerkin method (DGM) is one of the most potential high-order discretization method among the state-of-the-art methods, such as finite difference methods and finite volume method (FVM). This article is devoted to discussing the stability DGM coupled with continuous Galerkin (CG) in the simulation of INS problems. Ferrer and Willden[3] show the small time step instability in the simulation of unsteady Stokes and vortex flow with this scheme. A major preliminary of this work is that the procedure of solving the pressure during the splitting method is the most expensive stage, while approximation of the pressure with CG can significantly reduce computational cost compared to monolithic pure DG This is much more favorable for three-dimensional (3D) simulations. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. ‘‘Numerical results’’ section numerically investigate the small time and spatial stability through a serious of classical benchmark problems.

Methodology

Conclusion