Numerical upscaling for wave equations with time-dependent multiscale coefficients

TL;DR

This paper introduces a fully discrete multiscale method for wave equations with time-dependent coefficients, achieving optimal convergence and featuring an adaptive basis update strategy validated through numerical experiments.

Contribution

It develops a novel multiscale method that handles time-dependent coefficients without requiring spatial periodicity or scale separation, including an adaptive basis update mechanism.

Findings

Optimal convergence rates in space and time.

Effective adaptive basis update strategy.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we consider the classical wave equation with time-dependent, spatially multiscale coefficients. We propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with a backward Euler scheme in time. We show optimal convergence rates in space and time beyond the assumptions of spatial periodicity or scale separation of the coefficients. Further, we propose an adaptive update strategy for the time-dependent multiscale basis. Numerical experiments illustrate the theoretical results and showcase the practicability of the adaptive update strategy.