标题： 不可压缩欧拉方程的H(div)-协调无散度间断Galerkin方法与二阶显式Runge-Kutta格式的数值分析 显示英文标题

标题： Numerical analysis of an H(div)-conforming divergence-free discontinuous Galerkin method with a second-order explicit Runge-Kutta scheme for the incompressible Euler equations

摘要： 本文中，我们针对不可压缩Euler方程，提出了一个基于满足 \( H(\text{div}) \) -相容的无散度间断Galerkin (DG) 方法的误差分析，并结合了二阶显式Runge-Kutta (RK) 格式。 这项工作将二阶显式Runge-Kutta间断Galerkin (RKDG) 类型方法的误差分析扩展到了不可压缩流动问题，在其中精确的无散度假设引入了额外的分析挑战。 对于光滑解，我们在一个严格的CFL条件 $\tau \lesssim h^{4 / 3}$ 下，严格推导出多项式次数为 $k \geq 1$ 时 $O(h^{k+1 / 2}+\tau^2)$ 的先验误差估计，其中 $h$ 和 $\tau$ 分别是网格尺寸和时间步长。 对于线性多项式的情况，我们进一步研究现有的分析技术是否可以将这一严格的CFL条件放松到标准的CFL条件 $\tau \lesssim h$。 结果表明，精确的无散度约束阻止了这些技术的应用。我们推测，对于线性多项式，无法在标准CFL条件下推导出误差估计。 数值实验被进行，支持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： In this paper, we present an error analysis for an \( H(\text{div}) \)-conforming divergence-free discontinuous Galerkin (DG) method, combined with a second-order explicit Runge-Kutta (RK) scheme, for the incompressible Euler equations. This work extends the error analysis of the second-order explicit Runge-Kutta discontinuous Galerkin (RKDG) type methods to incompressible flows, in which the exactly divergence-free constraint introduces additional challenges to the analysis. For smooth solutions, we rigorously 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. For the case of linear polynomials, we further 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. Numerical experiments are conducted, supporting our analytical results and the conjecture for linear polynomials.