标题： 不可压缩流动的H(div)-协调无散度间断伽辽金方法与二阶显式龙格-库塔格式的数值分析 显示英文标题

标题： Numerical analysis of an H(div)-conforming divergence-free DG method with a second-order explicit Runge-Kutta scheme for incompressible flows

摘要： 最近，采用Runge-Kutta（RK）时间步进的H(div)协调DG型方法被广泛用于模拟高雷诺数流动，其中对流项显式处理。尽管RKDG方法的分析技术已得到充分发展，但由于无散度约束的存在，将其扩展到不可压缩流动问题上非常非平凡，关键在于分析对流项。我们忽略粘性效应，并针对不可压缩欧拉方程组，对结合二阶显式RK方案的H(div)协调无散度DG方法进行了误差分析。在严格的CFL条件 $\tau \lesssim h^{4 / 3}$ 下，对于多项式次数为 $k \geq 1$ 的情况，我们得到了 $O(h^{k+1 / 2}+\tau^2)$ 的先验误差估计，其中 $h$ 和 $\tau$ 分别为网格尺寸和时间步长，假设精确解足够光滑。 对于线性多项式的情形，我们研究了现有的分析技术是否可以将严格的CFL条件放松为标准的CFL条件 $\tau \lesssim h$。结果表明，无散度约束阻止了这些技术的应用。我们推测，在标准CFL条件下无法推导出线性多项式的误差估计。最后，我们指出基于我们的分析框架，分析结果可轻松推广至高网格雷诺数下的Navier-Stokes方程，其中粘性和对流项均显式处理。 数值实验结果支持了我们的分析结果以及关于线性多项式的猜想。 显示英文摘要

摘要： Recently, H(div)-conforming DG type methods coupled with Runge-Kutta (RK) time stepping have been widely employed for simulating high Reynolds number flows, with the convective terms treated explicitly. Although the analysis techniques of RKDG methods were well developed, the extension to incompressible flows is highly nontrivial due to the exactly divergence-free constraint, where the key lies in analyzing the convective terms. We neglect viscosity effects, and conduct an error analysis for an H(div)-conforming divergence-free DG method combined with a second-order explicit RK scheme, for the incompressible Euler equations. We derive an a priori error estimate of $O(h^{k+1 / 2}+\tau^2)$ under a restrictive CFL condition $\tau \lesssim h^{4 / 3}$ for polynomials of degree $k \geq 1$, where $h$ and $\tau$ are the mesh size and time step size, respectively, assuming that the exact solution is smooth. For the case of linear polynomials, we investigate whether existing analytical techniques can relax the restrictive CFL condition to a standard CFL condition $\tau \lesssim h$. It is demonstrated that the exactly divergence-free constraint prevents the application of these techniques. We conjecture that the error estimates for linear polynomials cannot be derived under a standard CFL condition. Finally, we mention that based on our analytical framework, our analytical results will be readily extended to the Navier-Stokes equations at high mesh Reynolds number, with the viscous and convective terms treated explicitly. Numerical experiments are conducted, supporting our analytical results and the conjecture for linear polynomials.