A Discontinuous Galerkin Consistent Splitting Method for the Incompressible Navier-Stokes Equations

TL;DR

This paper introduces a discontinuous Galerkin method for incompressible Navier-Stokes equations that enforces divergence-free velocity, removes pressure boundary layers, and achieves high accuracy with efficient decoupled computations.

Contribution

It develops a novel DG discretization of a consistent splitting scheme that implicitly enforces divergence-free velocity and simplifies pressure boundary conditions.

Findings

Achieves optimal convergence rates in space and time.

Demonstrates applicability to complex flow problems like flow around a cylinder and Taylor-Green vortex.

Removes pressure boundary layers and improves mass conservation.

Abstract

This work presents the discontinuous Galerkin discretization of the consistent splitting scheme proposed by Liu [J. Liu, J. Comp. Phys., 228(19), 2009]. The method enforces the divergence-free constraint implicitly, removing velocity--pressure compatibility conditions and eliminating pressure boundary layers. Consistent boundary conditions are imposed, also for settings with open and traction boundaries. Hence, accuracy in time is no longer limited by a splitting error. The symmetric interior penalty Galerkin method is used for second spatial derivatives. The convective term is treated in a semi-implicit manner, which relaxes the CFL restriction of explicit schemes while avoiding the need to solve nonlinear systems required by fully implicit formulations. For improved mass conservation, Leray projection is combined with divergence and normal continuity penalty terms. By selecting appropriate fluxes for both the divergence of the velocity field and the divergence of the convective operator, the consistent pressure boundary condition can be shown to reduce to contributions arising solely from the acceleration and the viscous term for the $L^2$ discretization. Per time step, the decoupled nature of the scheme with respect to the velocity and pressure fields leads to a single pressure Poisson equation followed by a single vector-valued convection-diffusion-reaction equation. We verify optimal convergence rates of the method in both space and time and demonstrate compatibility with higher-order time integration schemes. A series of numerical experiments, including the two-dimensional flow around a cylinder benchmark and the three-dimensional Taylor--Green vortex problem, verify the applicability to practically relevant flow problems.