Patch dynamics for macroscale modelling in one dimension

Abstract

We discuss efficient macroscale modelling of microscale systems using patch dynamics. This pilot study effectively homogenises microscale varying diffusion in one dimension. The `equation free' approach requires that the microscale model be solved only on small spatial patches. Suitable boundary conditions ensure that these patches are well coupled. By centre manifold theory, an emergent closed model exists on the macroscale. Patch dynamics systematically approximates this macroscale model. The modelling is readily adaptable to higher dimensions and to reaction-diffusion equations. References

• Dror Givon, Raz Kupferman, and Andrew Stuart. Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity, 17(6):R55, 2004. doi:10.1088/0951-7715/17/6/R01
• Francis B. Hildebrand. An introduction to numerical analysis. Dover, 1987.
• James M. Hyman. Patch dynamics for multiscale problems. Comput. Sci. Eng., 7(3):47--53, 2005. http://link.aip.org/link/?CSX/7/47/1
• Ioannis G. Kevrekidis and Giovanni Samaey. Equation-free multiscale computation: Algorithms and applications. Annu. Rev. Phys. Chem., 60(1):321--344, 2009. doi:10.1146/annurev.physchem.59.032607.093610
• S. Knapek. Matrix-dependent multigrid homogenization for diffusion problems. SIAM J. Sci. Comput., 20(2):515--533, 1998. doi:10.1137/S1064827596304848
• J. Moller, O. Runborg, P. G. Kevrekidis, K. Lust, and I. G. Kevrekidis. Equation-free, effective computation for discrete systems: a time stepper based approach. Int. J. Bifurcat. Chaos, 15(3):975--996, 2005. doi:10.1142/S0218127405012399
• A. J. Roberts. Holistic discretization ensures fidelity to {B}urgers' equation. Appl. Numer. Math., 37(3):371 -- 396, 2001. doi:10.1016/S0168-9274(00)00053-2
• A. J. Roberts. A holistic finite difference approach models linear dynamics consistently. Math. Comput., 72:247--262, 2003. doi:10.1090/S0025-5718-02-01448-5
• A. J. Roberts and I. G. Kevrekidis. General tooth boundary conditions for equation free modeling. SIAM J. Sci. Comput., 29(4):1495--1510, 2007. doi:10.1137/060654554
• A. J. Roberts, Tony MacKenzie, and J. E. Bunder. Accurate macroscale modelling of spatial dimensions. Technical report, http://arxiv.org/abs/1103.1187, 2011
• Olof Runborg, Constantinos Theodoropoulos, and Ioannis G Kevrekidis. Effective bifurcation analysis: a time-stepper-based approach. Nonlinearity, 15(2):491. doi:10.1088/0951-7715/15/2/314
• Giovanni Samaey, Anthony J. Roberts, and Ioannis G. Kevrekidis. Equation-free computation: an overview of patch dynamics. In Jacob Fish, editor, Bridging the Scales in Science and Engineering, pages 216--246. New York, Oxford University Press, 2010.
• Giovanni Samaey, Dirk Roose, and Ioannis G. Kevrekidis. Combining the gap-tooth scheme with projective integration: Patch dynamics. In Bjorn Engquist, Olof Runborg, and Per Lotstedt, editors, Multiscale Methods in Science and Engineering, volume 44 of Lecture Notes in Computational Science and Engineering, pages 225--239. Springer Berlin Heidelberg, 2005. doi:10.1007/3-540-26444-2\protect \global \let \OT1\textunderscore \unhbox \voidb@x \kern .06em\vbox {\hrule width.3em}\OT1\textunderscore 12