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

2025-12-05

AI Summary

This work addresses three key bottlenecks in numerically solving the incompressible Navier–Stokes equations: (1) conventional projection methods explicitly violate the divergence-free constraint; (2) velocity–pressure compatibility conditions limit achievable accuracy; and (3) inconsistencies arise in pressure boundary layers and under open or traction boundary conditions. We propose a novel implicit fractional-step scheme based on discontinuous Galerkin (DG) discretization. By incorporating a Leray projection and penalty terms enforcing divergence- and normal-continuity constraints, the method implicitly enforces solenoidality, thereby eliminating pressure boundary layers and compatibility restrictions. Viscous terms are discretized via symmetric interior penalty DG, while convective terms are treated semi-implicitly, ensuring high-order spatiotemporal convergence and strict mass conservation. The scheme supports arbitrary high-order time integration and rigorously satisfies consistency for open and traction boundaries. Optimal convergence rates and robustness are verified on benchmark problems including flow past a cylinder and the Taylor–Green vortex.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Develops a discontinuous Galerkin method for incompressible Navier-Stokes equations.
• Enforces divergence-free constraint implicitly to remove compatibility conditions.
• Achieves decoupled velocity-pressure solving with optimal convergence rates.

Innovation

• Discontinuous Galerkin discretization of consistent splitting scheme
• Semi-implicit convective term relaxes CFL restriction
• Decoupled scheme solves pressure Poisson then convection-diffusion equation