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

TL;DR

This paper extends the equation-free patch dynamics framework to two-dimensional problems on non-rectangular domains by formulating an explicit patch dynamics scheme in generalized curvilinear coordinates. By mapping the physical domain to a rectangular computational domain, it enables non-uniform grid clustering and curved patch geometries while solving a coarse evolution equation via coarse projective integration. The approach is validated on a convection–diffusion–reaction system and on non-axisymmetric diffusion in an annulus, showing high accuracy and grid-independence, with stretched grids (λ≈0.1) yielding notable improvements in high-gradient regions. The results demonstrate the method’s ability to bridge microscale and macroscale behaviour in complex geometries, offering a practical tool for multiscale modelling in physics and engineering.

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.

Figures (8)

• Figure 1: 9-point stencil
• Figure 2: Diffusion and convection dominated areas are represented with respect to the Peclet number.
• Figure 3: Left: Physical Plane & Right: Computational Plane
• Figure 4: Contour plots of the percentage erros of the patch dynamics solutions with congregated grids 11$\times$11, $Nt=2000$ and 21$\times$21, $Nt=8500$ at time $t=1$ are shown here, with $\lambda=0.1$ in both of the solutions.
• Figure 5: Contour plots of the percentage erros of the patch dynamics solutions with congregated grids 11$\times$11, $Nt=2000$ and 21$\times$21, $Nt=8500$ at time $t=1$ are shown here, with $\lambda=0.1$ in both of the solutions.