Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations

Abstract

This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation Forecast step: &(3ṽⁿ⁺²−4vⁿ⁺¹+vⁿ)/2Δt+ṽⁿ⁺² · ∇ ṽⁿ⁺² − νΔṽⁿ⁺² + ∇ qⁿ⁺²=f(x) notag &∇ · ṽⁿ⁺² = 0 notag Analysis step: &(3vⁿ⁺²−3ṽⁿ⁺²)/2Δt−χI_H(u(tⁿ⁺²)−vⁿ⁺²)=0. notag If I_H=I_H², the analysis step can be made explicit, taking the form vⁿ⁺²=ṽⁿ⁺²+2Δtχ/(3+2Δtχ)I_H(uⁿ⁺²−ṽⁿ⁺²). notag This implies the analysis step has the stability property of an implicit step and lower complexity than an explicit analysis step. Stability and error estimates for the BDF2 scheme are presented along with their proofs. Numerical experiments are conducted to assess the performance of BDF2 modular assimilation algorithm. The results of the experiments support the conclusion that modular data assimilation has comparable accuracy to standard, fully coupled data assimilation while greatly reducing computational complexity and cost.

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BibTeX RIS

BibTeX

@misc{yang2026higher,
  title = {{Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations}},
  author = {Troy Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.19508},
  eprint = {2606.19508},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.19508},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations
AU  - Troy Yang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2606.19508
UR  - https://arxiv.org/abs/2606.19508
AB  - This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation Forecast step: &(3ṽ^n+2−4v^n+1+v^n)/2Δt+ṽ^n+2 · ∇ ṽ^n+2 − νΔṽ^n+2 + ∇ q^n+2=f(x) notag &∇ · ṽ^n+2 = 0 notag Analysis step: &(3v^n+2−3ṽ^n+2)/2Δt−χI_H(u(t^n+2)−v^n+2)=0. notag If I_H=I_H^2, the analysis step can be made explicit, taking the form v^n+2=ṽ^n+2+2Δtχ/(3+2Δtχ)I_H(u^n+2−ṽ^n+2). notag This implies the analysis step has the stability property of an implicit step and lower complexity than an explicit analysis step. Stability and error estimates for the BDF2 scheme are presented along with their proofs. Numerical experiments are conducted to assess the performance of BDF2 modular assimilation algorithm. The results of the experiments support the conclusion that modular data assimilation has comparable accuracy to standard, fully coupled data assimilation while greatly reducing computational complexity and cost.
ER  -

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