Parallel in time partially explicit splitting scheme for high contrast linear multiscale diffusion problems

TL;DR

This paper introduces a parallel, partially explicit splitting scheme for high-contrast multiscale diffusion problems, enabling efficient and accurate simulations by decoupling spatial and temporal complexities and leveraging parallel computation.

Contribution

It proposes a novel parallel algorithm using a partially explicit splitting scheme that maintains stability and accuracy for high-contrast multiscale diffusion problems, independent of contrast levels.

Findings

Achieves high numerical accuracy for high-contrast problems.

Converges in a small number of iterations.

Iteration count remains stable with increasing coarse intervals.

Abstract

Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively captured, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider the all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.