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.

翻译：本文提出了Liu [J. Liu, J. Comp. Phys., 228(19), 2009]提出的一致分裂格式的离散伽辽金离散化方法。该方法隐式施加散度自由约束，消除了速度-压力相容性条件并去除了压力边界层。对于开放边界和牵引边界等情形，也施加了一致的边界条件。因此，时间精度不再受分裂误差的限制。空间二阶导数采用对称内罚伽辽金方法处理。对流项采用半隐式处理，既放宽了显式格式的CFL限制，又避免了全隐式格式需要求解非线性系统的问题。为改善质量守恒，将Leray投影与散度和法向连续罚项相结合。通过为速度场的散度与对流项的散度选择适当的通量，可证明对于$L^2$离散化，一致压力边界条件可简化为仅由加速度项和粘性项产生的贡献。每步时间积分中，速度场与压力场的解耦特性使得仅需先求解一个压力泊松方程，再求解一个向量值的对流-扩散-反应方程。我们验证了该方法在空间和时间上的最优收敛阶，并证明了其与高阶时间积分格式的兼容性。一系列数值实验（包括二维圆柱绕流基准问题和三维Taylor-Green涡问题）验证了该方法在实际流动问题中的适用性。