An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form

TL;DR

This paper introduces an equation-free multiscale method for efficiently approximating solutions to second order hyperbolic equations with rapidly oscillating coefficients, avoiding expensive fine-scale simulations.

Contribution

It develops a novel equation-free homogenization approach with proven convergence rates for multiscale hyperbolic problems in non-divergence form.

Findings

Proven convergence rates in periodic settings

Numerical validation in 1D and 2D cases

Cost-effective approximation of multiscale hyperbolic solutions

Abstract

The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. These small scales influence the large scale behavior, hence their effects should be accurately modelled in a numerical simulation. A direct numerical simulation is prohibitively expensive since a minimum of two points per wavelength are needed to resolve the small scales. A multiscale method, under the equation free methodology, is proposed to approximate the coarse scale behaviour of the exact solution at a cost independent of the small scales in the problem. We prove convergence rates for the upscaled quantities in one as well as in multi-dimensional periodic settings. Moreover, numerical results in one and two dimensions are provided to support the theory.