A Numerical Study of Continuous Data Assimilation for the 2D-NS Equations Using Nodal Points
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This thesis conducts a number of numerical experiments using massively parallel GPU computations to study a new continuous data assimilation algorithm. We test the algorithm on two-dimensional incompressible fluid flows given by the Navier-Stokes equations. In this context, observations of the Eulerian velocity field given at a finite resolution of nodal points in space may be used to recover the exact velocity field over time. We also consider nodal measurements of the vorticity field and stream function. The main difference between this new algorithm and previous continuous data assimilation methods is the inclusion of a relaxation parameter μ that controls the rate at which the approximate solution is forced toward the observational measurements. If μ is too small, the approximate solution obtained by data assimilation may not converge to the reference solution; however, if μ is too large then high frequency spill-over from the observations may contaminate the approximate solution. Our focus is on the resolution of the nodal points necessary for the algorithm to recover the exact velocity field and how best to choose the parameter μ.