  The patch dynamics scheme in equation-free multiscale modelling can
efficiently predict the macroscopic behaviours by simulating the microscale
problem in a fraction of the space-time domain. The patch dynamics schemes
developed so far, are mainly on rectangular domains with uniform grids and
uniform rectangular patches. In real-life problems where the geometry of the
domain is not regular or simple, rectangular and uniform grids or patches may
not be useful. To address this kind of complexity, the concept of a generalized
curvilinear coordinate system is used. An explicit representation of a patch
dynamics scheme on a generalized curvilinear coordinate system in a
two-dimensional domain is proposed for evolution equations. It has been applied
to unsteady convection-diffusion-reaction (CDR) problems. The robustness of the
scheme on the generalized curvilinear coordinate system is assessed through
numerical test cases. Firstly, a convection-dominated CDR equation is
considered, featuring high gradient regions in some part of the domain, for
which stretched grids with non-uniform patch sizes are employed. Secondly, a
non-axisymmetric diffusion equation is examined in an annulus region, where the
patches have non-rectangular shapes. The results obtained demonstrate excellent
agreement with the analytical solution or existing numerical solutions.
