The harmonic mean renormalises random diffusion across a spatial multigrid

Abstract

Most methods for modelling dynamics posit just two time scales: a fast and a slow scale. But many applications, such as the diffusion in a random media considered here, possess a wide variety of space-time scales. Consider the microscale diffusion on a one dimensional lattice with arbitrary diffusion coefficients between adjacent lattice points. I develop a slow manifold approach to model the diffusion, with some rigorous support, on a lattice that is coarser by a factor of four: the coarser scale effective diffusion coefficients are the harmonic mean of fine scale coefficients. Then iterating the analytic mapping of random diffusion from the finer grid to the coarser grid generates a hierarchy of models on a spatial multigrid across a wide range of space-time scales, all with rigorous support. The one step harmonic mean renormalises to harmonic means for the effective diffusion coefficients across the entire hierarchy. References Todd Arbogast and Kirsten J. Boyd. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal., 44:1150--1171, 2006. doi:10.1137/050631811. Achi Brandt. General highly accurate algebraic coarsening. Elect. Trans. Num. Anal., 10:1--20, 2000. http://www.emis.ams.org/journals/ETNA/vol.10.2000/pp1-20.dir/pp1-20.html. Achi Brandt. Multiscale scientific computation: review 2001. In T. F. Chan T. J. Barth and R. Haimes, editors, Multiscale and Multiresolution Methods: Theory and Applications, pages 1--96. Springer--Verlag, Heidelberg, 2001. Achi Brandt. Methods of systematic upscaling. Technical report, Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, March 2006. http://wisdomarchive.wisdom.weizmann.ac.il:81/archive/00000398/. William L. Briggs, Van Emden Henson, and Steve F. McCormick. A multigrid tutorial, second edition. SIAM, 2nd edition, 2001. J. Carr. Applications of centre manifold theory, volume 35 of Applied Math. Sci. Springer--Verlag, 1981. J. Dolbow, M. A. Khaleel, and J. Mitchell. Multiscale mathematics initiative: A roadmap. Report from the 3rd DoE workshop on multiscale mathematics. Technical report, Department of Energy, USA, http://www.sc.doe.gov/ascr/mics/amr, December 2004. Weinan E, Bjorn Engquist, Xiantao Li, Weiqing Ren, and Eric Vanden-Eijnden. The heterogeneous multiscale method: A review. Technical report, http://www.math.princeton.edu/multiscale/review.pdf, 2004. B. Engquist and P. E. Souganidis. Asymptotic and numerical homogenization. Acta Numerica, 17:147--190, 2008. Martin J. Gander and Andrew M. Stuart. Space-time continuous analysis of waveform relaxation for the heat equation. SIAM Journal on Scientific Computing, 19(6):2014--2031, 1998. Y. A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathematical Sciences. Springer--Verlag, 1995. Tony MacKenzie and A. J. Roberts. Holistic discretisation ensures fidelity to dynamics in two spatial dimensions. Technical report, http://arxiv.org/abs/0904.0855v1, 2009. G. A. Pavliotis and A. M. Stuart. Multiscale methods: averaging and homogenization, volume 53 of Texts in Applied Mathematics. Springer, 2008. A. J. Roberts. Simple and fast multigrid solution of Poisson's equation using diagonally oriented grids. ANZIAM J., 43(E):E1--E36, July 2001. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/465. A. J. Roberts. Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics. In Rowena Ball and Nail Akhmediev, editors, Nonlinear dynamics from lasers to butterflies, volume 1 of Lecture Notes in Complex Systems, chapter 7, pages 257--313. World Scientific, 2003. A. J. Roberts. Choose interelement coupling to preserve self-adjoint dynamics in multiscale modelling and computation. Technical report, http://arxiv.org/abs/0811.0688, 2008. A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. Physica A, 387:12--38, 2008. A. J. Roberts. Model dynamics across multiple length and time scales on a spatial multigrid. Multiscale Modeling and Simulation, 7(4):1525--1548, 2009. G. Samaey, I. G. Kevrekidis, and D. Roose. The gap-tooth scheme for homogenization problems. SIAM Multiscale Modeling and Simulation, 4:278--306, 2005. doi:10.1137/030602046. G. Samaey, A. J. Roberts, and I. G. Kevrekidis. Equation-free computation: an overview of patch dynamics, chapter 8, pages 216--246. Oxford University Press, 2010.