Temporal Splitting algorithms for non-stationary multiscale problems
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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" splitting of the solution space, one can achieve an efficient temporal splitting. We present a theoretical result, which was derived in our earlier paper and adopted in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.
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