Continuous and discrete data assimilation with noisy observations for the Rayleigh-Bénard convection: a computational study

• DOI: 10.1007/s10596-022-10180-4
• Publication Date: 2022-12-05T16:04:55Z
• Authors: Mohamad Abed El R...; Olivier Le Maître; Edriss S. Titi; Ibrahim Hoteit; Omar Knio
• Keywords: FOS: Computer and information sciences; 37 Earth Sciences; Bioengineering; Dynamical Systems (math.DS); [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA]; 01 natural sciences; Computational Engineering, Finance, and Science (cs.CE); Networking and Information Technology R&D (NITRD); 3701 Atmospheric Sciences; FOS: Mathematics; [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]; Networking and Information Technology R&D (NITRD); [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]; Mathematics - Dynamical Systems; Computer Science - Computational Engineering, Finance, and Science; [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]; 0105 earth and related environmental sciences

AbstractObtaining accurate high-resolution representations of model outputs is essential to describe the system dynamics. In general, however, only spatially- and temporally-coarse observations of the system states are available. These observations can also be corrupted by noise. Downscaling is a process/scheme in which one uses coarse scale observations to reconstruct the high-resolution solution of the system states. Continuous Data Assimilation (CDA) is a recently introduced downscaling algorithm that constructs an increasingly accurate representation of the system states by continuously nudging the large scales using the coarse observations. We introduce a Discrete Data Assimilation (DDA) algorithm as a downscaling algorithm based on CDA with discrete-in-time nudging. We then investigate the performance of the CDA and DDA algorithms for downscaling noisy observations of the Rayleigh-Bénard convection system in the chaotic regime. In this computational study, a set of noisy observations was generated by perturbing a reference solution with Gaussian noise before downscaling them. The downscaled fields are then assessed using various error- and ensemble-based skill scores. The CDA solution was shown to converge towards the reference solution faster than that of DDA but at the cost of a higher asymptotic error. The numerical results also suggest a quadratic relationship between the ℓ2 error and the noise level for both CDA and DDA. Cubic and quadratic dependences of the DDA and CDA expected errors on the spatial resolution of the observations were obtained, respectively.