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• Designed coupled implicit-explicit temporal and spatial splitting method. • Formulated conditions for stability and designed appropriate spaces. • Multiscale spaces with CEM-GMsFEM are used in constructing the splitting method. • Numerical results confirm our theoretical findings. Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is d t = H 2 / κ m a x , where κ m a x is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented. [ABSTRACT FROM AUTHOR]

• Design temporal splitting together with multiscale methods to reduce the computational cost. • Study temporal splitting and spatial multiscale decomposition to extract implicit and explicit modes. • Use of GMsFEM within temporal splitting methods appropriately. In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a "good" decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in [15] and adopt in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm. [ABSTRACT FROM AUTHOR]

Abstract: There are a number of modeling challenges posed by space weather simulations. Most of them arise from the multiscale and multiphysics aspects of the problem. The multiple scales dramatically increase the requirements, in terms of computational resources, because of the need of performing large scale simulations with the proper small-scales resolution. Lately, several suggestions have been made to overcome this difficulty by using various refinement methods which consist in splitting the domain into regions of different resolutions separated by well defined interfaces. The multiphysics issues are generally treated in a similar way: interfaces separate the regions where different equations are solved. This paper presents an innovative approach based on the coexistence of several levels of description, which differ by their resolutions or, potentially, by their physics. Instead of interacting through interfaces, these levels are entirely simulated and are interlocked over the complete extension of the overlap area. This scheme has been applied to a parallelized, two-dimensional, Implicit Moment Method Particle in Cell code in order to investigate its multiscale description capabilities. Simulations of magnetic reconnection and plasma expansion in vacuum are presented and possible implementation options for this scheme on very large systems are also discussed. [Copyright &y& Elsevier]

• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]