Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain

TL;DR

This paper introduces a wavelet-based edge multiscale parareal algorithm designed to efficiently solve subdiffusion equations with heterogeneous coefficients over long time periods, combining multiscale spatial methods with parallel-in-time strategies.

Contribution

The paper develops a novel algorithm that uncouples the temporal variable in subdiffusion problems, providing proven approximation properties and a new exponential summation scheme with complexity independent of final time.

Findings

Algorithm achieves convergence with respect to spatial and temporal discretization parameters.

Numerical tests confirm the theoretical convergence rates and efficiency.

Method effectively handles heterogeneity and nonlocality in subdiffusion equations.

Abstract

We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm, recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently solving subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the benefits of multiscale methods, which can handle heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Our algorithm overcomes the challenge posed by the nonlocality of the fractional derivative in previous parabolic problem work by constructing an auxiliary problem on each coarse temporal subdomain to completely uncouple the temporal variable. We prove the approximation properties of the correction operator and derive a new summation of exponential to generate a single-step time stepping scheme, with the number of terms of $\mathcal{O}(|\log{\tau_f}|^2)$ independent of the final time, where $\tau_f$ is the fine-scale time step size. We establish the convergence rate of our algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size, and the fine-scale time step size. Finally, we present several numerical tests that demonstrate the effectiveness of our algorithm and validate our theoretical results.