Modular data assimilation for flow prediction

This report develops several modular, 2-step realizations (inspired by Kalman filter algorithms) of nudging-based data assimilation $$Step \ 1 \quad \frac{\widetilde {v}^{n+1}-v^{n}}{k}+v^{n}\cdot \nabla \widetilde {v}^{n+1}-\nu \triangle \widetilde {v}^{n+1}+\nabla q^{n+1}=f(x)$$ $$\nabla \cdot \widetilde {v}^{n+1}=0$$ $$Step \ 2 \quad \frac{v^{n+1}-\widetilde {v}^{n+1}}{k}-\chi I_{H}(u(t^{n+1})-v^{n+1})=0.$$ Several variants of this algorithm are developed. Three main results are developed. The first is that if $I_{H}^{2}=I_{H}$, then Step 2 can be rewritten as the explicit step $$v^{n+1}=\widetilde {v}^{n+1}+\frac{k\chi }{1+k\chi }[I_{H}u(t^{n+1})-I_{H} \widetilde {v}^{n+1}].$$ This means Step 2 has the greater stability of an implicit update and the lesser complexity of an explicit analysis step. The second is that the basic result of nudging (that for $H$ small enough and $\chi$ large enough predictability horizons are infinite) holds for one variant of the modular algorithm. The third is that, for any $H>0$ and any $\chi>0$, one step of the modular algorithm decreases the next step's error and increases (an estimate of) predictability horizons. A method synthesizing assimilation with eddy viscosity models of turbulence is also presented. Numerical tests are given, confirming the effectiveness of the modular assimilation algorithm. The conclusion is that the modular, 2-step method overcomes many algorithmic inadequacies of standard nudging methods and retains a robust mathematical foundation.