Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

Joshua Hudson; Michael Jolly

doi:10.3934/jcd.2019006

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June 2019, 6(1): 131-145. doi: 10.3934/jcd.2019006

Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

Joshua Hudson ^1,2, and Michael Jolly ^3,,

1.	University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21055, USA
2.	Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723-6099, USA
3.	Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

^* Corresponding author: Michael Jolly

Published August 2019

We study the computational efficiency of several nudging data assimilation algorithms for the 2D magnetohydrodynamic equations, using varying amounts and types of data. We find that the algorithms work with much less resolution in the data than required by the rigorous estimates in [7]. We also test other abridged nudging algorithms to which the analytic techniques in [7] do not seem to apply. These latter tests indicate, in particular, that velocity data alone is sufficient for synchronization with a chaotic reference solution, while magnetic data alone is not. We demonstrate that a new nonlinear nudging algorithm, which is adaptive in both time and space, synchronizes at a super exponential rate.

Keywords: Magnetohydrodynamic equations, data assimilation, synchronization.

Mathematics Subject Classification: Primary: 34D06, 76W05.

Citation: Joshua Hudson, Michael Jolly. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations. Journal of Computational Dynamics, 2019, 6 (1) : 131-145. doi: 10.3934/jcd.2019006

References:

[1]	D. A. Albanez, H. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164. doi: 10.3233/ASY-151351.
[2]	M. U. Altaf, E. S. Titi, T. Gebrael, O. M. Knio, L. Zhao, M. F. McCabe and I. Hoteit, Downscaling the 2D Bénard convection equations using continuous data assimilation, Comput. Geosci., 21 (2017), 393-410. doi: 10.1007/s10596-017-9619-2.
[3]	A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.
[4]	A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579.
[5]	H. Bessaih, E. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753. doi: 10.1088/0951-7715/28/3/729.
[6]	A. Biswas, C. Foias, C. F. Mondaini and E. S. Titi, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326. doi: 10.1016/j.anihpc.2018.05.004.
[7]	A. Biswas, J. Hudson, A. Larios and Y. Pei, Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.
[8]	K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet and B. Brown, Dedalus: Flexible framework for spectrally solving differential equations, Astrophysics Source Code Library, 2016.
[9]	E. Celik, E. Olson and E. S. Titi, Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142. doi: 10.1137/18M1218480.
[10]	J. Charney, M. Halem and R. Jastrow, Use of incomplete historical data to infer the present state of the atmosphere, Journal of the Atmospheric Sciences, 26 (1969), 1160-1163. doi: 10.1175/1520-0469(1969)026<1160:UOIHDT>2.0.CO;2.
[11]	R. Dascaliuc, C. Foias and M. Jolly, Relations between energy and enstrophy on the global attractor of the 2-d Navier-Stokes equations, J. Dynamics Differential Equations, 17 (2005), 643-736. doi: 10.1007/s10884-005-8269-6.
[12]	D. Drake, C. Kletzing, F. Skiff, G. Howes and S. Vincena, Design and use of an Elsässer probe for analysis of Alfén wave field according to wave direction, The Review of Scientific Instruments, 82 (2011), 103505.
[13]	G. Evensen, Data Assimilation, 2nd edition, Springer-Verlag, Berlin, 2009, The ensemble Kalman filter. doi: 10.1007/978-3-642-03711-5.
[14]	A. Farhat, H. Johnston, M. Jolly and E. S. Titi, Assimilation of nearly turbulent Rayleigh-Bénard flow through vorticity or local circulation measurements: a computational study, J. Sci. Comput., 77 (2018), 1519-1533. doi: 10.1007/s10915-018-0686-x.
[15]	A. Farhat, M. S. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66. doi: 10.1016/j.physd.2015.03.011.
[16]	A. Farhat, E. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23. doi: 10.1007/s00021-015-0225-6.
[17]	A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, Journal of Mathematical Analysis and Applications, 438 (2016), 492-506. doi: 10.1016/j.jmaa.2016.01.072.
[18]	A. Farhat, E. Lunasin and E. S. Titi, On the Charney conjecture of data assimilation employing temperature measurements alone: The paradigm of 3d planetary geostrophic model, arXiv: 1608.04770.
[19]	A. Farhat, E. Lunasin and E. S. Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. Nonlinear Sci., 27 (2017), 1065-1087. doi: 10.1007/s00332-017-9360-y.
[20]	M. Gesho, E. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110. doi: 10.4208/cicp.060515.161115a.
[21]	M. Ghil, M. Halem and R. Atlas, Time-continuous assimilation of remote-sounding data and its effect on weather forecasting, Mon. Wea. Rev., 107 (1978), 140-171.
[22]	K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D, 240 (2011), 1416-1425. doi: 10.1016/j.physd.2011.04.021.
[23]	A. Jackson, A. R. T. Jonkers and M. R. Walker, Four centuries of geomagnetic secular variation from historical records, Phil. Trans. R. Soc. Lond., 358 (2000), 957-990. doi: 10.1098/rsta.2000.0569.
[24]	M. S. Jolly, V. R. Martinez and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17 (2017), 167-192.
[25]	M. S. Jolly, T. Sadigov and E. S. Titi, Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case, Nonlinear Anal. Real World Appl., 36 (2017), 287-317. doi: 10.1016/j.nonrwa.2017.01.010.
[26]	A. R. T. Jonkers, A. Jackson and A. Murray, Four centuries of geomagnetic data from historical records, Rev. Geophys., 41.
[27]	A. Larios and Y. Pei, Nonlinear continuous data assimilation, arXiv: 1703.03546.
[28]	D. Leunberger, An introduction to observers, IEEE Trans. Automat. Control, 16 (1971), 596-602. doi: 10.1109/TAC.1971.1099826.
[29]	P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328. doi: 10.1088/0951-7715/29/4/1292.
[30]	E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence, J. Statist. Phys., 113 (2003), 799–840, Progress in statistical hydrodynamics (Santa Fe, NM, 2002). doi: 10.1023/A:1027312703252.
[31]	S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015. doi: 10.1017/CBO9781107706804.
[32]	F. E. Thau, Observing the state of non-linear dynamic systems, Int. J. Control, 17 (1973), 471-479. doi: 10.1080/00207177308932395.

show all references

References:

[1]	D. A. Albanez, H. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model, Asymptotic Anal., 97 (2016), 139-164. doi: 10.3233/ASY-151351.
[2]	M. U. Altaf, E. S. Titi, T. Gebrael, O. M. Knio, L. Zhao, M. F. McCabe and I. Hoteit, Downscaling the 2D Bénard convection equations using continuous data assimilation, Comput. Geosci., 21 (2017), 393-410. doi: 10.1007/s10596-017-9619-2.
[3]	A. Azouani, E. Olson and E. S. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277-304. doi: 10.1007/s00332-013-9189-y.
[4]	A. Azouani and E. S. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters—a reaction-diffusion paradigm, Evol. Equ. Control Theory, 3 (2014), 579-594. doi: 10.3934/eect.2014.3.579.
[5]	H. Bessaih, E. Olson and E. S. Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity, 28 (2015), 729-753. doi: 10.1088/0951-7715/28/3/729.
[6]	A. Biswas, C. Foias, C. F. Mondaini and E. S. Titi, Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 295-326. doi: 10.1016/j.anihpc.2018.05.004.
[7]	A. Biswas, J. Hudson, A. Larios and Y. Pei, Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43.
[8]	K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet and B. Brown, Dedalus: Flexible framework for spectrally solving differential equations, Astrophysics Source Code Library, 2016.
[9]	E. Celik, E. Olson and E. S. Titi, Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm, SIAM J. Appl. Dyn. Syst., 18 (2019), 1118-1142. doi: 10.1137/18M1218480.
[10]	J. Charney, M. Halem and R. Jastrow, Use of incomplete historical data to infer the present state of the atmosphere, Journal of the Atmospheric Sciences, 26 (1969), 1160-1163. doi: 10.1175/1520-0469(1969)026<1160:UOIHDT>2.0.CO;2.
[11]	R. Dascaliuc, C. Foias and M. Jolly, Relations between energy and enstrophy on the global attractor of the 2-d Navier-Stokes equations, J. Dynamics Differential Equations, 17 (2005), 643-736. doi: 10.1007/s10884-005-8269-6.
[12]	D. Drake, C. Kletzing, F. Skiff, G. Howes and S. Vincena, Design and use of an Elsässer probe for analysis of Alfén wave field according to wave direction, The Review of Scientific Instruments, 82 (2011), 103505.
[13]	G. Evensen, Data Assimilation, 2nd edition, Springer-Verlag, Berlin, 2009, The ensemble Kalman filter. doi: 10.1007/978-3-642-03711-5.
[14]	A. Farhat, H. Johnston, M. Jolly and E. S. Titi, Assimilation of nearly turbulent Rayleigh-Bénard flow through vorticity or local circulation measurements: a computational study, J. Sci. Comput., 77 (2018), 1519-1533. doi: 10.1007/s10915-018-0686-x.
[15]	A. Farhat, M. S. Jolly and E. S. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D, 303 (2015), 59-66. doi: 10.1016/j.physd.2015.03.011.
[16]	A. Farhat, E. Lunasin and E. S. Titi, Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18 (2016), 1-23. doi: 10.1007/s00021-015-0225-6.
[17]	A. Farhat, E. Lunasin and E. S. Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, Journal of Mathematical Analysis and Applications, 438 (2016), 492-506. doi: 10.1016/j.jmaa.2016.01.072.
[18]	A. Farhat, E. Lunasin and E. S. Titi, On the Charney conjecture of data assimilation employing temperature measurements alone: The paradigm of 3d planetary geostrophic model, arXiv: 1608.04770.
[19]	A. Farhat, E. Lunasin and E. S. Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, J. Nonlinear Sci., 27 (2017), 1065-1087. doi: 10.1007/s00332-017-9360-y.
[20]	M. Gesho, E. Olson and E. S. Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier-Stokes equations, Commun. Comput. Phys., 19 (2016), 1094-1110. doi: 10.4208/cicp.060515.161115a.
[21]	M. Ghil, M. Halem and R. Atlas, Time-continuous assimilation of remote-sounding data and its effect on weather forecasting, Mon. Wea. Rev., 107 (1978), 140-171.
[22]	K. Hayden, E. Olson and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D, 240 (2011), 1416-1425. doi: 10.1016/j.physd.2011.04.021.
[23]	A. Jackson, A. R. T. Jonkers and M. R. Walker, Four centuries of geomagnetic secular variation from historical records, Phil. Trans. R. Soc. Lond., 358 (2000), 957-990. doi: 10.1098/rsta.2000.0569.
[24]	M. S. Jolly, V. R. Martinez and E. S. Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17 (2017), 167-192.
[25]	M. S. Jolly, T. Sadigov and E. S. Titi, Determining form and data assimilation algorithm for weakly damped and driven Korteweg–de Vries equation—Fourier modes case, Nonlinear Anal. Real World Appl., 36 (2017), 287-317. doi: 10.1016/j.nonrwa.2017.01.010.
[26]	A. R. T. Jonkers, A. Jackson and A. Murray, Four centuries of geomagnetic data from historical records, Rev. Geophys., 41.
[27]	A. Larios and Y. Pei, Nonlinear continuous data assimilation, arXiv: 1703.03546.
[28]	D. Leunberger, An introduction to observers, IEEE Trans. Automat. Control, 16 (1971), 596-602. doi: 10.1109/TAC.1971.1099826.
[29]	P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328. doi: 10.1088/0951-7715/29/4/1292.
[30]	E. Olson and E. S. Titi, Determining modes for continuous data assimilation in 2D turbulence, J. Statist. Phys., 113 (2003), 799–840, Progress in statistical hydrodynamics (Santa Fe, NM, 2002). doi: 10.1023/A:1027312703252.
[31]	S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, New York, 2015. doi: 10.1017/CBO9781107706804.
[32]	F. E. Thau, Observing the state of non-linear dynamic systems, Int. J. Control, 17 (1973), 471-479. doi: 10.1080/00207177308932395.

Figure 1. Properties of the reference solution. In (a) and (b) the evolution of the $ L^2 $ norm of the reference solution computed with $ 256^2 $ resolution is shown for $ t\in[0,100] $ and $ t\in[10,792.92] $. A comparison with $ 512^2 $ resolution is in (a). Plots of the $ H^1 $ semi-norm vs the $ L^2 $ norm of the solutions for $ t\in[10,90] $ with $ 256^2 $ and $ 512^2 $ resolutions are in (c) and (d) respectively

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Figure 2. Contour lines of the curl of the computed reference solution at time $ t = 729.92 $

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Figure 3. Dependence of the error (14) on $ \mu $ and $ N $. The solutions were computed over the time interval $ [729.9,734.9] $, and the error is at time $ t = 734.9 $

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Figure 4. Convergence results for Algorithms 2.2, 2.3, and 2.4, with damping $ \mu = 20 $

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Figure 5. Convergence results for Algorithms 4.1-4.4 with $ \mu = 20 $

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Figure 6. The evolution of the $ L^2 $ error is shown for solutions of nonlinear modifications of Algorithm 2.2. Each simulation was performed with $ N = 32 $

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Figure 7. The evolution of the $ L^2 $ error is shown for solutions of Algorithm 2.2 when subject to simulated measurement noise

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