A Generalized Curvilinear Coordinate system-based Patch Dynamics Scheme in Equation-free Multiscale Modelling

Abstract: 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.