Date of this Version
C. Victor, Sweeping Probe Data Assimilation: Time-Dependent Grid Interpolation for a Reaction-Diffusion Equation, undergraduate thesis, Univ. of Nebraska - Lincoln, Lincoln, 2018.
This research project applied the Azouani-Olson-Titi data assimilation algorithm to the 1D Chafee-Infante/Allen-Cahn equation in order to investigate the potential for different grid configurations. It was discovered that for a wide range of parameter values convergence rates could be significantly increased by replacing a uniform static grid of data assimilation points by a moving cluster of data assimilation points. In addition to this, this study found a heuristic argument solving the inverse problem of determining the minimum length scale given a viscosity value, via analyzing computationally the minimum number of data assimilation nodes required for convergence. All simulations were conducted using a semi-implicit convex-splitting numerical scheme for the time stepping and spatial derivatives were approximated using second-order finite difference approximations.