Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates
Fully discrete numerical schemes of a data assimilation algorithm: uniform-in-time error estimates
复制标题
数据同化算法的完全离散数值方案：时间均匀误差估计
DOI：
10.1093/imanum/drz043
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发表时间：
2019
影响因子：
2.1
通讯作者：
E. Titi
中科院分区：
文献类型：
--
作者：
H. Ibdah;Cecilia F. Mondaini;E. Titi
我们的目标是近似的参考速度场求解二维Navier-Stokes方程（NSE）在其初始条件的情况下，利用空间离散测量的字段，可在一个粗略的规模，并在时间上连续。通过数值离散化降尺度数据同化算法获得的近似。时间离散是基于半隐式和全隐式欧拉格式，而空间离散（可以在任意尺度上进行，而不管测量的空间分辨率如何）是基于谱伽辽金方法。这两个完全离散的算法被证明是无条件稳定的，相对于时间步长的大小，时间步长的数量和Galerkin模式的数量。此外，在L^2 $和H^1 $范数下，得到了近似解和参考解之间的显式、时间上一致的误差估计。值得注意的是，二维NSE，无滑移Dirichlet或周期性边界条件，在这项工作中使用的范例。在整体存在唯一性的假设下，本文给出的完整分析可以推广到其他二维和三维耗散系统。
Our aim is to approximate a reference velocity field solving the two-dimensional Navier–Stokes equations (NSE) in the absence of its initial condition by utilizing spatially discrete measurements of that field, available at a coarse scale, and continuous in time. The approximation is obtained via numerically discretizing a downscaling data assimilation algorithm. Time discretization is based on semiimplicit and fully implicit Euler schemes, while spatial discretization (which can be done at an arbitrary scale regardless of the spatial resolution of the measurements) is based on a spectral Galerkin method. The two fully discrete algorithms are shown to be unconditionally stable, with respect to the size of the time step, the number of time steps and the number of Galerkin modes. Moreover, explicit, uniform-in-time error estimates between the approximation and the reference solution are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional NSE, subject to the no-slip Dirichlet or periodic boundary conditions, are used in this work as a paradigm. The complete analysis that is presented here can be extended to other two- and three-dimensional dissipative systems under the assumption of global existence and uniqueness.