论文标题
关于Evolution Navier-Stokes方程的时间数值离散的二阶收敛
On the convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations
论文作者
论文摘要
我们证明了某些二阶数值方法与局部能量不平等的navier-stokes方程的弱解的收敛性，因此在Scheffer和Caffarelli-Kohn-Nirenberg的意义上适合。更确切地说，我们在三个空间量量中对待空间周期性案例，我们考虑了一个完全离散化，其中使用经典的曲柄 - 尼科尔森方法（$θ$ -thod，带有$θ= 1/2 $）来离散时间变量，而在太空变量中，我们考虑使用有限元。对流术语以几种隐式，半义和明确的方式离散。特别是，我们专注于证明离散解决方案朝着弱解决方案（满足精确的局部能量平衡）的（可能有条件的）收敛，而没有对极限问题的额外规律性假设。我们没有证明融合的顺序，但是我们的分析确定了一些数值方案，这些方案也提供了在三个空间维度中“物理相关”解决方案存在的其他证据。
We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, we treat the space-periodic case in three space-dimensions and we consider a full discretization in which the the classical Crank-Nicolson method ($θ$-method with $θ=1/2$) is used to discretize the time variable, while in the space variables we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions towards weak solutions (satisfying a precise local energy balance), without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes providing also alternate proofs of existence of "physically relevant" solutions in three space dimensions.
