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首页> 外文期刊>Journal of Computational Physics >On the stability of projection methods for the incompressible Navier–Stokes equations based on high-order discontinuous Galerkin discretizations
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On the stability of projection methods for the incompressible Navier–Stokes equations based on high-order discontinuous Galerkin discretizations

机译：基于高阶不连续的Galerkin决定的不可压缩Navier-Stokes方程投影方法的稳定性

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AbstractThe present paper deals with the numerical solution of the incompressible Navier–Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for small time step sizes. Since the critical time step size depends on the viscosity and the spatial resolution, these instabilities limit the robustness of the Navier–Stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf–sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf–sup stability explicitly.]]>
机译：<！[cdata [ Abstract 本文涉及使用高阶不连续Galerkin（DG）方法的不可压缩Navier-Stokes方程的数值解决方案，以便在空间中离散化。对于应用于双分裂投影方法的DG方法，最近报告了少时阶梯尺寸的稳定性。由于临界时间步长取决于粘度和空间分辨率，因此这些不稳定性限制了Navier-Stokes求解器的鲁棒性，例如在粗略空间分辨率和小粘度的复杂工程应用的情况下。通过数值调查，我们提供了这些不稳定性与速度分歧项的不连续的Galerkin配方和耦合速度和压力的压力梯度术语有关。为了获得稳定且稳健的方法，需要通过这些术语的各个部分集成，以获得稳定和稳健的方法。由于中间速度字段不满足对速度规定的边界条件，因此从双分割方案的对流步骤导出一致的边界条件，以确保关于时间离散化的高阶精度。这种新的配方在等级和混合多项式近似的小时间步骤的极限下稳定。虽然双分裂方案本身包括INF-SUP稳定贡献，但我们证明了杂散的压力振荡出现了相等的多项式和小型时间步长，突出了明确考虑INF-SUP稳定性的必要性。 ]]>

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