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Analysis of the Picard-Newton Iteration for the Navier-Stokes Equations: Global Stability and Quadratic Convergence

https://doi.org/10.1007/s10915-025-02946-6

Pollock, Sara; Rebholz, Leo_G; Tu, Xuemin; Xiao, Mengying (May 2025, Journal of Scientific Computing)

Abstract We analyze and test a simple-to-implement two-step iteration for the incompressible Navier-Stokes equations that consists of first applying the Picard iteration and then applying the Newton iteration to the Picard output. We prove that this composition of Picard and Newton converges quadratically, and our analysis (which covers both the unique solution and non-unique solution cases) also suggests that this solver has a larger convergence basin than usual Newton because of the improved stability properties of Picard-Newton over Newton. Numerical tests show that Picard-Newton converges more reliably for higher Reynolds numbers and worse initial conditions than Picard and Newton iterations. We also consider enhancing the Picard step with Anderson acceleration (AA), and find that the AAPicard-Newton iteration has even better convergence properties on several benchmark test problems.
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Performance assessment of the maximum likelihood ensemble filter and the ensemble Kalman filters for nonlinear problems

https://doi.org/10.1007/s40687-022-00359-7

Wang, Yijun; Zupanski, Milija; Tu, Xuemin; Gao, Xinfeng (December 2022, Research in the Mathematical Sciences)

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Implicit particle filtering via a bank of nonlinear Kalman filters

https://doi.org/10.1016/j.automatica.2022.110469

Askari, Iman; Haile, Mulugeta A.; Tu, Xuemin; Fang, Huazhen (November 2022, Automatica)

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A direct filter method for parameter estimation

https://doi.org/10.1016/j.jcp.2019.108871

Archibald, Richard; Bao, Feng; Tu, Xuemin (December 2019, Journal of Computational Physics)
null (Ed.)
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A BDDC algorithm for the Stokes problem with weak Galerkin discretizations

https://doi.org/10.1016/j.camwa.2018.04.024

Tu, Xuemin; Wang, Bin (July 2018, Computers & Mathematics with Applications)

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Projected Shadowing-Based Data Assimilation

https://doi.org/10.1137/17M1141163

de Leeuw, Bart; Dubinkina, Svetlana; Frank, Jason; Steyer, Andrew; Tu, Xuemin; Vleck, Erik Van (January 2018, SIAM Journal on Applied Dynamical Systems)

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