Continuous data assimilation for the magnetohydrodynamic equations in 2D using one component of the velocity and magnetic fields

Abstract

We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in $L^2$ and $H^1$ norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.

34 pages, submitted to SIAM Journal on Mathematical Analysis

References in corpus (8)

• Continuous Data Assimilation for the 2D Bénard Convection through Velocity Measurements Alone
• Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model
• Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field
• Continuous Data Assimilation with Stochastically Noisy Data
• A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations
• Continuous Data Assimilation for a 2D Bénard Convection System through Horizontal Velocity Measurements Alone
• Nonlinear Continuous Data Assimilation
• Postprocessing Galerkin method applied to a data assimilation algorithm: a uniform in time error estimate

Cited by in corpus (2)

• Concurrent multi-parameter learning demonstrated on the Kuramoto-Sivashinsky equation
• Sensitivity Analysis for the 2D Navier-Stokes Equations with Applications to Continuous Data Assimilation