A high-order multi-scale method and its error analysis for nonlocal-in-time fractionally damped wave problems in composite materials
Abstract
Significant spatial heterogeneity and nonlocal history-dependent attenuation jointly govern wave propagation in composite materials, making full-scale direct numerical simulation (DNS) computationally prohibitive. This study proposed a high-order multi-scale (HOMS) method for nonlocal-in-time fractionally damped wave problems with highly oscillatory and discontinuous material coefficients. Using a multi-scale asymptotic expansion approach, the macroscopic homogenized equation, and first- and second-order correction terms are derived to construct the high-accuracy HOMS solution. Moreover, a local pointwise error analysis identifies the leading residual eliminated by this HOMS solution and validates its local physical balance preservation. Then, a global energy-norm error estimate establishes the O(ε)-order convergence rate for the HOMS solution. Subsequently, an efficient spatiotemporal multi-scale algorithm integrating the finite element and finite difference methods is designed, which employs a novel fully discrete L2−1σ-Newmark scheme, and corresponding unconditional shifted-energy stability and explicit convergence rates are also obtained for the macroscopic homogenized displacement and velocity, respectively. Both two- and three-dimensional examples are presented to demonstrate the accuracy, efficiency, reduced cost, and robustness of the proposed HOMS method. This work provides an effective multi-scale tool for nonlocal-in-time wave simulation in engineering applications.
Related
- Stabilized SQP Method for Convex Piecewise Linear-Quadratic Composite Optimization Problems
- C1-estimates in nonlocal parabolic problems
- Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
- Nonlocal problems for Laplace equation in Bochner spaces
- Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
BibTeX
@misc{wang2026highorder,
title = {{A high-order multi-scale method and its error analysis for nonlocal-in-time fractionally damped wave problems in composite materials}},
author = {Gehao Wang and Hao Dong and Yufeng Nie and Junzhi Cui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29451},
eprint = {2610.29451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29451},
}
RIS
TY - UNPB
TI - A high-order multi-scale method and its error analysis for nonlocal-in-time fractionally damped wave problems in composite materials
AU - Gehao Wang
AU - Hao Dong
AU - Yufeng Nie
AU - Junzhi Cui
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29451
UR - https://arxiv.org/abs/2610.29451
AB - Significant spatial heterogeneity and nonlocal history-dependent attenuation jointly govern wave propagation in composite materials, making full-scale direct numerical simulation (DNS) computationally prohibitive. This study proposed a high-order multi-scale (HOMS) method for nonlocal-in-time fractionally damped wave problems with highly oscillatory and discontinuous material coefficients. Using a multi-scale asymptotic expansion approach, the macroscopic homogenized equation, and first- and second-order correction terms are derived to construct the high-accuracy HOMS solution. Moreover, a local pointwise error analysis identifies the leading residual eliminated by this HOMS solution and validates its local physical balance preservation. Then, a global energy-norm error estimate establishes the O(ε)-order convergence rate for the HOMS solution. Subsequently, an efficient spatiotemporal multi-scale algorithm integrating the finite element and finite difference methods is designed, which employs a novel fully discrete L2−1_σ-Newmark scheme, and corresponding unconditional shifted-energy stability and explicit convergence rates are also obtained for the macroscopic homogenized displacement and velocity, respectively. Both two- and three-dimensional examples are presented to demonstrate the accuracy, efficiency, reduced cost, and robustness of the proposed HOMS method. This work provides an effective multi-scale tool for nonlocal-in-time wave simulation in engineering applications.
ER -