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

TL;DR

This work develops a high-order discontinuous Galerkin discretization of the consistent splitting scheme for the incompressible Navier–Stokes equations, enabling implicit divergence-free enforcement and consistent boundary treatment on open boundaries. The method decouples velocity and pressure updates via a pressure Poisson equation and a momentum equation, employing a semi-implicit convective term and Leray projection for improved mass conservation. It demonstrates optimal convergence in space and time, and proves robust on classical benchmarks like flow around a cylinder and the 3D Taylor–Green vortex, including higher-order time integrators that maintain stability at large CFL numbers. The approach offers a scalable, accurate alternative to fully coupled solvers while avoiding pressure boundary layers inherent to standard projection methods."

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.

Figures (6)

• Figure 1: Temporal convergence of velocity (left) and pressure (right) in the Taylor--Green vortex benchmark as obtained via the consistent splitting scheme for BDF-1 to BDF-4.
• Figure 2: Spatial convergence of the consistent splitting scheme with different polynomial degrees.
• Figure 3: Flow around a cylinder benchmark: computational grid after 2 uniform refinements (top) and horizontal component $u_x$ of the velocity vector with developed vortex shedding (bottom).
• Figure 4: Kinetic energy, molecular dissipation and numerical dissipation of the consistent splitting method with 21 million velocity DoFs in comparison to literature results taken from Brachet1983Gassner2013Fehn2018b. Values from Gassner2013 are only available for $t\in[0,10]$. Lower numerical dissipation means more accurate, limited by the fixed spatiotemporal resolution chosen.
• Figure 5: Comparison between the solutions of the coupled solution approach, the dual splitting scheme and the consistent splitting with 786432 velocity DoFs. All schemes yield similar results and compare well against the reference solution from Fehn2018b, which uses the dual splitting scheme with 3.221 billion DoFs. Lower numerical dissipation means more accurate, limited by the fixed spatiotemporal resolution chosen.

Theorems & Definitions (3)

• Remark
• Remark
• Remark