Equation-free patch scheme for efficient computational homogenisation via self-adjoint coupling

TL;DR

This paper introduces a novel equation-free patch scheme with self-adjoint coupling conditions that preserve key symmetries, ensuring accurate and efficient macroscale modeling of microscale systems across multiple dimensions.

Contribution

The authors develop and analyze a new patch coupling scheme that maintains invariance and symmetry properties, enhancing the accuracy and consistency of multiscale simulations.

Findings

Scheme preserves conservation laws and symmetries.

Spectral and algebraic analyses improve accuracy.

Proven macroscale dynamics match microscale models.

Abstract

Equation-free macroscale modelling is a systematic and rigorous computational methodology for efficiently predicting the dynamics of a microscale system at a desired macroscale system level. In this scheme, the given microscale model is computed in small patches spread across the space-time domain, with patch coupling conditions bridging the unsimulated space. For accurate simulations, care must be taken in designing the patch coupling conditions. Here we construct novel coupling conditions which preserve translational invariance, rotational invariance, and self-adjoint symmetry, thus guaranteeing that conservation laws associated with these symmetries are preserved in the macroscale simulation. Spectral and algebraic analyses of the proposed scheme in both one and two dimensions reveal mechanisms for further improving the accuracy of the simulations. Consistency of the patch scheme's macroscale dynamics with the original microscale model is proved. This new self-adjoint patch scheme provides an efficient, flexible, and accurate computational homogenisation in a wide range of multiscale scenarios of interest to scientists and engineers.