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

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.

High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier-Stokes equations

“…In our previous work [49], the penalty terms have been applied in the projection equation (40). However, since we evaluate the continuity penalty operator also on boundary faces in the present work, the penalty terms are evaluated in a postprocessing step, equation (42). Adding the penalty terms to the projection equation (40) would prevent to achieve high-order temporal accuracy since prescribing the boundary condition g u for the intermediate velocity ûh would be inconsistent.…”