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

TL;DR

This paper tackles efficient long-time simulation of time-fractional diffusion with heterogeneous coefficients by marrying a spatial multiscale reduction (WEMsFEM) with a time-parareal strategy. A key novelty is reducing the history term via a summation of exponentials, decoupling temporal evolution and bounding memory growth independently of the final time. The authors establish stability and convergence analyses for the multiscale SOE-scheme and for the overall WEMP algorithm, and they demonstrate rapid convergence (often within three iterations) in numerical experiments across smooth and rough sources and long time horizons. The resulting framework significantly lowers computational and storage costs while preserving accuracy, making it suitable for large-scale, long-time simulations in heterogeneous media.

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 $\mathcal{O}(|\log{τ_f}|^2)$ independent of the final time, where $τ_f$ 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.

Figures (8)

• Figure 1: Illustration of a coarse neighborhood and coarse element.
• Figure 2: $\alpha=0.1$, $\epsilon^*=0.5$.
• Figure 3: $\alpha=0.9$, $\epsilon^*=0.5$.
• Figure 4: $\alpha=0.9$, $\epsilon^*=0.5$.
• Figure 5: $\alpha=0.1$, $\epsilon^*=0.5$.

Theorems & Definitions (26)

• Lemma 2.1
• proof
• Lemma 4.1
• Theorem 4.1: Approximation by exponential sums for long time
• proof
• Corollary 4.1
• proof
• Remark 4.1: Alternative approximation by exponential sums
• Lemma 4.2
• proof