13 research outputs found
Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations
Based on a previously introduced downscaling data assimilation algorithm,
which employs a nudging term to synchronize the coarse mesh spatial scales, we
construct a determining map for recovering the full trajectories from their
corresponding coarse mesh spatial trajectories, and investigate its properties.
This map is then used to develop a downscaling data assimilation scheme for
statistical solutions of the two-dimensional Navier-Stokes equations, where the
coarse mesh spatial statistics of the system is obtained from discrete spatial
measurements. As a corollary, we deduce that statistical solutions for the
Navier-Stokes equations are determined by their coarse mesh spatial
distributions. Notably, we present our results in the context of the
Navier-Stokes equations; however, the tools are general enough to be
implemented for other dissipative evolution equations
Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm
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 and 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
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Postprocessing Galerkin method applied to a data assimilation algorithm: a uniform in time error estimate
We apply the Postprocessing Galerkin method to a recently introduced
continuous data assimilation (downscaling) algorithm for obtaining a numerical
approximation of the solution of the two-dimensional Navier-Stokes equations
corresponding to given measurements from a coarse spatial mesh. Under suitable
conditions on the relaxation (nudging) parameter, the resolution of the coarse
spatial mesh and the resolution of the numerical scheme, we obtain uniform in
time estimates for the error between the numerical approximation given by the
Postprocessing Galerkin method and the reference solution corresponding to the
measurements. Our results are valid for a large class of interpolant operators,
including low Fourier modes and local averages over finite volume elements.
Notably, we use here the 2D Navier-Stokes equations as a paradigm, but our
results apply equally to other evolution equations, such as the Boussinesq
system of Benard convection and other oceanic and atmospheric circulation
models
Postprocessing Galerkin method applied to a data assimilation algorithm: a uniform in time error estimate
We apply the Postprocessing Galerkin method to a recently introduced
continuous data assimilation (downscaling) algorithm for obtaining a numerical
approximation of the solution of the two-dimensional Navier-Stokes equations
corresponding to given measurements from a coarse spatial mesh. Under suitable
conditions on the relaxation (nudging) parameter, the resolution of the coarse
spatial mesh and the resolution of the numerical scheme, we obtain uniform in
time estimates for the error between the numerical approximation given by the
Postprocessing Galerkin method and the reference solution corresponding to the
measurements. Our results are valid for a large class of interpolant operators,
including low Fourier modes and local averages over finite volume elements.
Notably, we use here the 2D Navier-Stokes equations as a paradigm, but our
results apply equally to other evolution equations, such as the Boussinesq
system of Benard convection and other oceanic and atmospheric circulation
models
Recommended from our members
Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm
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 and 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