Modular data assimilation for flow prediction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912455623442432 |
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| author | Çıbık, Aytekin Fang, Rui Layton, William |
| author_facet | Çıbık, Aytekin Fang, Rui Layton, William |
| contents | This report develops several modular, 2-step realizations (inspired by Kalman filter algorithms) of nudging-based data assimilation $$Step \ 1 \quad \frac{\widetilde {v}^{n+1}-v^{n}}{k}+v^{n}\cdot \nabla \widetilde {v}^{n+1}-ν\triangle \widetilde {v}^{n+1}+\nabla q^{n+1}=f(x)$$ $$\nabla \cdot \widetilde {v}^{n+1}=0$$ $$Step \ 2 \quad \frac{v^{n+1}-\widetilde {v}^{n+1}}{k}-χI_{H}(u(t^{n+1})-v^{n+1})=0.$$ 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 $$v^{n+1}=\widetilde {v}^{n+1}+\frac{kχ}{1+kχ}[I_{H}u(t^{n+1})-I_{H} \widetilde {v}^{n+1}].$$ 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 $χ$ large enough predictability horizons are infinite) holds for one variant of the modular algorithm. The third is that, for any $H>0$ and any $χ>0$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular data assimilation for flow prediction Çıbık, Aytekin Fang, Rui Layton, William Numerical Analysis 65M12, 37L99, 76D55 This report develops several modular, 2-step realizations (inspired by Kalman filter algorithms) of nudging-based data assimilation $$Step \ 1 \quad \frac{\widetilde {v}^{n+1}-v^{n}}{k}+v^{n}\cdot \nabla \widetilde {v}^{n+1}-ν\triangle \widetilde {v}^{n+1}+\nabla q^{n+1}=f(x)$$ $$\nabla \cdot \widetilde {v}^{n+1}=0$$ $$Step \ 2 \quad \frac{v^{n+1}-\widetilde {v}^{n+1}}{k}-χI_{H}(u(t^{n+1})-v^{n+1})=0.$$ 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 $$v^{n+1}=\widetilde {v}^{n+1}+\frac{kχ}{1+kχ}[I_{H}u(t^{n+1})-I_{H} \widetilde {v}^{n+1}].$$ 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 $χ$ large enough predictability horizons are infinite) holds for one variant of the modular algorithm. The third is that, for any $H>0$ and any $χ>0$, 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. |
| title | Modular data assimilation for flow prediction |
| topic | Numerical Analysis 65M12, 37L99, 76D55 |
| url | https://arxiv.org/abs/2506.19002 |