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Symplectic integration of PDEs using Clebsch variables
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,,
University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21055, USA
Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723-6099, USA
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
* Corresponding author: Michael Jolly
Published August 2019
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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
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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 $
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 $
Figure 4. Convergence results for Algorithms 2.2, 2.3, and 2.4, with damping $ \mu = 20 $
Figure 5. Convergence results for Algorithms 4.1-4.4 with $ \mu = 20 $
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 $
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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Joshua Hudson Michael Jolly