Temporal Splitting algorithms for non-stationary multiscale problems

TL;DR

This paper explores temporal splitting algorithms for multiscale, non-stationary problems, leveraging multiscale spatial decompositions to improve efficiency in solving time-dependent PDEs.

Contribution

It introduces a novel temporal splitting approach based on multiscale spatial approximations, enhancing efficiency for non-stationary multiscale problems.

Findings

Numerical results demonstrate improved computational efficiency.

Theoretical results support the effectiveness of the proposed splitting.

Multiscale decomposition enables better implicit-explicit discretization.

Abstract

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 our earlier work and adopt it in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm.