Generalized Patch Dynamics Scheme in Equation-free Multiscale Modelling

TL;DR

This paper addresses the challenge of extracting macroscopic dynamics when only a known microscopic evolution exists. It introduces Generalized Patch Dynamics (GPD), which distributes gap-tooth timesteps across three time scales—micro, meso, and macro—through two schemes (Type-I and Type-II) and an optional PI-GPD extension, to reduce extrapolation error and improve efficiency. Through a 1D unsteady reaction-diffusion test, GPD demonstrates higher accuracy and faster computation than the usual patch dynamics (UPD), and can overcome UPD divergence for long extrapolations by using nonuniform GTT distributions and shorter mesoscopic steps. The work positions GPD as a unifying, flexible framework linking UPD, HMM, FLAVORS, VSHMM, and BA, with potential applicability to stiff and oscillatory multiscale problems.

Abstract

There is a class of problems that exhibit smooth behavior on macroscopic scales, where only a microscopic evolution law is known. Patch dynamics scheme of `equation-free multiscale modelling' is one of the techniques, which aims to extract the macroscopic information using such known time-dependent microscopic model simulation in patches (which is a fraction of the space-time domain) that reduces the computational complexity. Here, extrapolation time step has an important role to reduce the error at macroscopic level. In this study, a generalized patch dynamics (GPD) scheme is proposed by distributing the gap-tooth timesteppers (GTTs) within each long (macroscopic) time step. This distribution is done in two ways, namely, GPD schemes of type-I and type-II. The proposed GPD scheme is based on three different time scales namely, micro, meso and macro to predict the system level behaviours. The GPD scheme of both types are capable of providing better accuracy with less computation time compared to the usual patch dynamics (UPD) scheme. The physical behaviours of the problems can be more appropriately addressed by the GPD scheme as one may use a non-uniform (variable) distribution of gap-tooth timesteppers (GTTs), as well as the extrapolation times based on the physics of the problem. Where the UPD scheme fails to converge for a long extrapolation time, both types of GPD schemes can be successfully applied. The whole method has been analyzed successfully for the one-dimensional reaction-diffusion problem.

Figures (14)

• Figure 1: A schematic description of the gap-tooth timestepper (GTT).
• Figure 1: A schematic description of the GPD scheme of Type-I.
• Figure 1: Comparison between the analytical solution and the GPD solutions of type-I of the effective macroscopic equation of \ref{['eqn:22']} at times $t=0.01, t=0.1$ and $t=0.2$ are shown in this figure.
• Figure 2: A schematic description of the usual patch dynamics scheme.
• Figure 2: A schematic description of the GPD scheme of Type-II.