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Suitable reduced order models (ROMs) are computationally efficient tools in characterizing key dynamical and statistical features of nature. In this paper, a systematic multiscale stochastic ROM framework is developed for complex systems with strong chaotic or turbulent behavior. The new ROMs fundamentally differ from the traditional Galerkin ROM (G-ROM) or those deterministic closure ROMs that aim to minimize path-wise errors and apply mainly to laminar systems. Here, the new ROM focuses on recovering the large-scale dynamics to the maximum extent. At the same time, it also exploits cheap but effective conditionally linear functions as the closure terms to capture the statistical features of the medium-scale variables and their feedback to the large scales. The new ROM mimics the model structure of many complex systems, allowing it to be more physically explainable. In addition, physics constraints of energy-conserving nonlinearities are incorporated into the new ROM. One unique feature of the resulting ROM is that it facilitates an efficient and accurate scheme for nonlinear data assimilation, the solution of which is provided by closed analytic formulae that do not require ensemble methods. Such an analytical solvable data assimilation solution significantly accelerates computational efficiency. It allows the new ROM to avoid many potential numerical and sampling issues in recovering the unobserved states from partial observations. The closure term calibration is efficient and systematic via explicit mathematical formulae. The new ROM framework is applied to complex nonlinear systems in which the intrinsic turbulent behavior is either triggered by external random forcing or deterministic nonlinearity. The out-of-sample numerical results show that the new ROM significantly outperforms the G-ROM in both scenarios regarding reproducing the dynamical and statistical features and recovering unobserved states via the efficient data assimilation scheme. • A systematic multiscale stochastic ROM framework is developed. • The new ROM can be applied to complex systems with strong chaotic behavior. • Physics constraints are incorporated into the new ROM. • Closed analytic formulae for data assimilation is provided by the new ROM. • It facilitates an efficient and accurate scheme for nonlinear data assimilation. [ABSTRACT FROM AUTHOR]

The HLL solver for the generalized Riemann problem (HLL-GRP solver) is attractive because it is easy to implement for 1D hyperbolic problems and performs well. Based on the rotational invariance of multidimensional magnetohydrodynamics (MHD) equations, this paper provides a means of implementing the 1D HLL-GRP solver into the multidimensional ideal MHD simulations within the framework of the finite volume method. Furthermore, the Godunov-Powell source term and the locally divergence-free WLS-ENO reconstruction are adopted to constrain the divergence of the magnetic field. To maintain positive density and pressure, a positivity-preserving (PP) limiter is performed on the reconstructed polynomials of density and pressure. Some comparisons with other methods are listed in 1D, 2D and 3D MHD benchmark cases. The numerical results demonstrate that the scheme is stable and can well control the divergence of the magnetic field. [ABSTRACT FROM AUTHOR]