An Equation-Free Approach for Second Order Multiscale Hyperbolic Problems in Non-Divergence Form
For researchers in numerical homogenization, this work provides a novel method for a class of hyperbolic problems where direct simulation is expensive, but the contribution is incremental as it extends existing equation-free techniques to a specific problem type.
The paper proposes an equation-free multiscale method for second-order hyperbolic equations in non-divergence form with rapidly oscillating coefficients, achieving coarse-scale approximations at a cost independent of small scales. Convergence rates are proven and numerical results in 1D and 2D support the theory.
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.