Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time | Paper Citation Record

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This paper cites Tracking the model: data a s- similation by artiﬁcial neural network.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Tracking the model: data a s- similation by artiﬁcial neural network

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This paper cites A machine learning aug- mented data assimilation method for high resolution observations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time A machine learning aug- mented data assimilation method for high resolution observations

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This paper cites Data Assimilation for Geo- physical Fluids.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Data Assimilation for Geo- physical Fluids

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This paper cites Physics-informe d neural networks for multiphysics data assimilation with applica- tion to subsurface transport.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Physics-informe d neural networks for multiphysics data assimilation with applica- tion to subsurface transport

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This paper cites Deep data assimilation: integrat- ing deep learning with data assimilation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Deep data assimilation: integrat- ing deep learning with data assimilation

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This paper cites Data Assimilation with Deep Neural Nets Informed by Nudging.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Data Assimilation with Deep Neural Nets Informed by Nudging

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This paper cites A novel neural network training fra mework with data assimilation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time A novel neural network training fra mework with data assimilation

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This paper cites Iterative methods for data a ssimilation for Burgers’ equation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Iterative methods for data a ssimilation for Burgers’ equation

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This paper cites A nudging-based data assimilation method for oceanographuc problems: the back and forth nudging (BFN) algor ithm.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time A nudging-based data assimilation method for oceanographuc problems: the back and forth nudging (BFN) algor ithm

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This paper cites Unsteady adjoint method for the optimal con trol of ad- vection and Burgers’ equation using high order spectral diﬀerenc e method.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Unsteady adjoint method for the optimal con trol of ad- vection and Burgers’ equation using high order spectral diﬀerenc e method

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This paper cites The back and forth nudging algorithm for da ta assimi- lation problems: theoretical results on transport equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time The back and forth nudging algorithm for da ta assimi- lation problems: theoretical results on transport equations

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This paper cites An evolution of the back and forth nudg- ing for geophysical data assimilation: application to Burgers equatio n and comparison.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time An evolution of the back and forth nudg- ing for geophysical data assimilation: application to Burgers equatio n and comparison

Reference 12

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Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Numerical aspects of large-tim e optimal control of Burgers’ equation

Reference 13

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This paper cites Filtered gradient algorithms for inverse des ign problems of one-dimensional Burgers’ equation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Filtered gradient algorithms for inverse des ign problems of one-dimensional Burgers’ equation

Reference 14

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This paper cites Special issue on in- verse problems in geosciences.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Special issue on in- verse problems in geosciences

Reference 15

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This paper cites Contaminant source identiﬁcation in ac- quifers: a critical view.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Contaminant source identiﬁcation in ac- quifers: a critical view

Reference 16

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This paper cites Non-point contamin ant source identiﬁcation in an acquifer using the ensemble smoother with multiple d ata assimilation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Non-point contamin ant source identiﬁcation in an acquifer using the ensemble smoother with multiple d ata assimilation

Reference 17

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This paper cites Properties of advection algo rithms in the context of variational data assimilation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Properties of advection algo rithms in the context of variational data assimilation

Reference 18

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Observation 1966be4b-d6d0-4a87-8a9c-ca1afb641ef8 · outbound

This paper cites Data assimilation in 2D nonlinear advection diﬀusion eq ua- tions, using an explicit stabilized leapfrog scheme run backwrd in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Data assimilation in 2D nonlinear advection diﬀusion eq ua- tions, using an explicit stabilized leapfrog scheme run backwrd in time

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This paper cites Opportunistic capacity and error exponent regions for compound channel with feedback.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Opportunistic capacity and error exponent regions for compound channel with feedback

Reference 21

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Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Diﬀerence Methods for Initial Value P roblems

Reference 22

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This paper cites Compensating operators and stable backward in t ime march- ing in nonlinear parabolic equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Compensating operators and stable backward in t ime march- ing in nonlinear parabolic equations

Reference 23

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This paper cites Stable explicit time-marching in well-posed or ill-posed non- linear parabolic equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stable explicit time-marching in well-posed or ill-posed non- linear parabolic equations

Reference 24

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This paper cites Stable explicit marching scheme in ill-posed time-reve rsed vis- cous wave equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stable explicit marching scheme in ill-posed time-reve rsed vis- cous wave equations

Reference 25

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This paper cites Stabilized Richardson leapfrog scheme in explicit ste pwise computation of forward or backward nonlinear parabolic equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stabilized Richardson leapfrog scheme in explicit ste pwise computation of forward or backward nonlinear parabolic equations

Reference 26

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This paper cites Stable explicit stepwise marching scheme in ill- posed t ime- reversed 2D Burgers’ equation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stable explicit stepwise marching scheme in ill- posed t ime- reversed 2D Burgers’ equation

Reference 28

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This paper cites Computing ill-posed time-reversed 2D Navier-Stok es equa- tions, using a stabilized explicit ﬁnite diﬀerence scheme marching back ward in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Computing ill-posed time-reversed 2D Navier-Stok es equa- tions, using a stabilized explicit ﬁnite diﬀerence scheme marching back ward in time

Reference 29

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This paper cites Stabilized leapfrog scheme run backward in time, and the explicit O(∆ t)2 stepwise computation of ill-posed time- reversed 2D Navier-Stokes equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stabilized leapfrog scheme run backward in time, and the explicit O(∆ t)2 stepwise computation of ill-posed time- reversed 2D Navier-Stokes equations

Reference 30

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This paper cites Finite diﬀerence schemes for incompressible ﬂow based on local pressure boundary conditions.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Finite diﬀerence schemes for incompressible ﬂow based on local pressure boundary conditions

Reference 31

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Observation a8123f5c-e5d4-419b-950b-725436f6e10a · outbound

This paper cites Solution of 2D Navier-Stokes equat ion by cou- pled ﬁnite diﬀerence-dual reciprocity boundary element method.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Solution of 2D Navier-Stokes equat ion by cou- pled ﬁnite diﬀerence-dual reciprocity boundary element method

Reference 32

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This paper cites Non-Standard and Improperly Posed P roblems.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Non-Standard and Improperly Posed P roblems

Reference 33

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This paper cites Some remarks on ill-posed problems for viscous ﬂuids.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Some remarks on ill-posed problems for viscous ﬂuids

Reference 34

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This paper cites Logarithmic convexity and other techniques applied t o prob- lems in continuum mechanics.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Logarithmic convexity and other techniques applied t o prob- lems in continuum mechanics

Reference 35

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Observation 7e80c4c7-4229-4d84-bbcf-2a4a8793045f · outbound

This paper cites On the stability of solutions of the Navier-S tokes equations backward in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time On the stability of solutions of the Navier-S tokes equations backward in time

Reference 36

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Observation 3d81a370-88aa-4095-9634-6ba9cbd8c1e5 · outbound

This paper cites Uniqueness and continuous dependence criteria for the Navier- Stokes equations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Uniqueness and continuous dependence criteria for the Navier- Stokes equations

Reference 37

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Observation c9ec6c4e-6b87-468b-ab99-4a07143343c4 · outbound

This paper cites Reconstructing the past from imprecise knowledg e of the present: Eﬀective non-uniqueness in solving parabolic equations ba ckward in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Reconstructing the past from imprecise knowledg e of the present: Eﬀective non-uniqueness in solving parabolic equations ba ckward in time

Reference 38

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Observation 2130a4bb-9a8a-4fcd-af77-9daff0a33a6f · outbound

This paper cites Computing small solutions of Burgers’ equation bac kwards in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Computing small solutions of Burgers’ equation bac kwards in time

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.802663Z

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Observation a636546c-d04d-4e33-b4e0-3d8185e99eff · outbound

This paper cites Stability estimates for Burger s-type equations backward in time.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stability estimates for Burger s-type equations backward in time

Reference 40

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Observation f921a477-097f-4168-9df4-bdddba347fbb · outbound

This paper cites An example of non-linear computational instability.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time An example of non-linear computational instability

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.782201Z

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Observation d1b94b57-f220-4dd3-9a8c-049c870290fa · outbound

This paper cites On the instability of Leapfrog and Crank-Nicolson a pproxima- tions of a nonlinear partial diﬀerential equation.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time On the instability of Leapfrog and Crank-Nicolson a pproxima- tions of a nonlinear partial diﬀerential equation

Reference 42

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Observation 9bc44036-f296-4f36-8cb6-214d68e3ce89 · outbound

This paper cites Studies in numerical nonlinear instability I.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Studies in numerical nonlinear instability I

Reference 43

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This paper cites Frequency ﬁlter for time integrations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Frequency ﬁlter for time integrations

Reference 44

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Observation 60b6f3ea-206c-41a2-a406-ecf29fb825eb · outbound

This paper cites A proposed modiﬁcation to the Robert-Asselin time ﬁlte r.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time A proposed modiﬁcation to the Robert-Asselin time ﬁlte r

Reference 45

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.743498Z

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No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation a2f9469d-6a20-40c2-8f12-362a6a47d792 · outbound

This paper cites The RA W ﬁlter: An improvement to the Robert-Asselin ﬁ lter in semi-implicit integrations.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time The RA W ﬁlter: An improvement to the Robert-Asselin ﬁ lter in semi-implicit integrations

Reference 46

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.732632Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 9c7d746d-2e73-4dfa-9d43-6f1148ae7470 · outbound

This paper cites The eﬀects of the RA W ﬁlter on th e climatology and forecast skill of the SPEEDY model.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time The eﬀects of the RA W ﬁlter on th e climatology and forecast skill of the SPEEDY model

Reference 47

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.722170Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

Observation 2f94cb8b-5bf9-4b40-93c3-da14fbd3a868 · outbound

This paper cites Stability analysis of the Crank-Nico lson- Leapfrog method with the Robert-Asselin-Williams time ﬁlter.

Data assimilation in 2D incompressible Navier-Stokes equations, using a stabilized explicit $O(\Delta t)^2$ leapfrog finite difference scheme run backward in time Stability analysis of the Crank-Nico lson- Leapfrog method with the Robert-Asselin-Williams time ﬁlter

Reference 48

Resolution: verified fuzzy; raw_fallback, observed 2026-08-12T15:12:52.712246Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-12T06:34:41.77262+00:00.

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