Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm
Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm
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数据同化算法的完全离散数值方案的统一时间误差估计
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
E. Titi
中科院分区：
文献类型：
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作者：
H. Ibdah;Cecilia F. Mondaini;E. Titi
作者：Ibdah，Hussain A; Mondaini，Cecilia F; Titi，Edriss S|摘要：我们认为完全离散的数值方案的降尺度数据同化算法的目的是近似的速度场的二维Navier-Stokes方程对应于给定的粗网格观测测量。时间离散是通过考虑半隐式和全隐式欧拉格式，空间离散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，得到了完全离散解和参考解之间的时间误差估计。值得注意的是，二维Navier-Stokes方程，无滑移Dirichlet或周期性的边界条件，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。
Author(s): Ibdah, Hussain A; Mondaini, Cecilia F; Titi, Edriss S | Abstract: We consider fully discrete numerical schemes for a downscaling data assimilation algorithm aimed at approximating the velocity field of the 2D Navier-Stokes equations corresponding to given coarse mesh observational measurements. The time discretization is done by considering semi- and fully-implicit Euler schemes, and the spatial discretization 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, number of time steps and the number of Galerkin modes. Moreover, explicit, uniform in time error estimates between the fully discrete solution and the reference solution corresponding to the observational coarse mesh measurements are obtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensional Navier-Stokes equations, 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.