  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.
