We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed fine-scale computation of such a problem requires discretization and solution of a large system of equations, and can be prohibitively time-consuming. To circumvent this difficulty, we propose an equation-free approach, where the fine-scale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarse-grained variables (observables) is evaluated during the fine-scale computation, and ``projective integration'' is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of fine-scale solutions with "correct" correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented.
Fourier series (based) multiscale method for computational analysis in science and engineering: IV. Fourier series multiscale solution for the convection-diffusion-reaction equation
Fourier series (based) multiscale method for computational analysis in science and engineering: III. Fourier series multiscale method for linear differential equation with constant coefficients
Fourier series (based) multiscale method for computational analysis in science and engineering: VII. Fourier series multiscale solution for elastic bending of beams on Pasternak foundations
An efficient solver based on low-rank approximation and Neumann matrix series for unsteady diffusion-type partial differential equations with random coefficients
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
Fourier series (based) multiscale method for computational analysis in science and engineering: VI. Fourier series multiscale solution for wave propagation in a beam with rectangular cross section
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the sixth paper, exact analysis of the wave propagation in a beam with rectangular cross section is extended to a thorough multiscale analysis for a system of completely coupled second order linear differential equations for modal functions, where general boundary conditions are prescribed. For this purpose, the modal function each is expressed as a linear combination of the corner function, the two boundary functions and the internal func…
Oversampling for the Multiscale Finite Element Method
Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster（应用与数值数学研究所，明斯特大学）；Institut für Mathematik, Humboldt-Universität zu Berlin（数学研究所，柏林洪堡大学）
Higher-Order Multiscale Computational Method for Multi-Continuum Problems in Highly Heterogeneous Media
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
An Exponential Time-Integrator Scheme for Steady and Unsteady Inviscid Flows
Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics and Statistics, Old Dominion University（老道明大学数学与统计系）；Department of Aerospace Engineering, University of Kansas（堪萨斯大学航空航天工程系）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
