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

Abstract

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.2010 Mathematics Subject Classification. Primary: 34D06, 76W05.

Continuous Data Assimilation For the 3D Ladyzhenskaya Model: Analysis and Computations

“…We demonstrate the efficacy of the algorithm by extensive computational studies. Numerical work with other fluid systems has shown that the nudging algorithm achieves synchronization with data that is much more coarse than required by the rigorous estimates [2,18,24,27]. We find this is also the case for the Ladyzhenskaya model with periodic boundary conditions, for which we achieve exponential convergence to machine precision with h ≈ 0.1.…”

Continuous Data Assimilation for the Three Dimensional Navier-Stokes Equations

“…the analytic Gevrey class) [2,9,20,22,23,24,25,41,42], as well as to more general situations such as discrete in time and errorcontaminated measurements and to statistical solutions [7,8,26]. This method has been shown to perform remarkably well in numerical simulations [3,21,30,32,33,38] and has recently been successfully implemented for the first time for efficient dynamical downscaling of a global atmospheric reanalysis [17]. Recent applications include its implementation in reduced order modeling (ROM) of turbulent flows to mitigate inaccuracies in ROM [49], and in inferring flow parameters and turbulence configurations [18,13].…”

Data assimilation for the Navier-Stokes equations using local observables

“…Conversely, when an algorithm works in practice, it suggests there might be some analysis to support it. Computational work has demonstrated that nudging over the entire computational domain works much better than required in the rigorous estimates [2,56,28,40,47,48] In our pseudospectral implementation, we have h = 2π/N, so strictly speaking Theorem 2.1 would require N ∼ G exp( √ N /2), which is far from obtainable. Yet our computational results are promising.…”