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1、PrinciplesofMultiscaleModelingWeinanEMay14,20112ContentsPrefacevii1Introduction11.1Examplesofmultiscaleproblems......................11.1.1Multiscaledataandtheirrepresentation..............21.1.2Di?erentialequationswithmultiscaledata.............21.1.3Di?erentialequationswithsmallparame
2、ters............41.2Multi-physicsproblems............................51.2.1Examplesofscale-dependentphenomena..............51.2.2De?cienciesofthetraditionalapproachestomodeling.......71.2.3Themulti-physicsmodelinghierarchy................101.3Analyticalmethods...........................
3、...121.4Numericalmethods..............................131.4.1Linearscalingalgorithms.......................131.4.2Sublinearscalingalgorithms.....................141.4.3TypeAandtypeBmultiscaleproblems..............141.4.4Concurrentvs.sequentialcoupling.................151.5Whatarethemain
4、challenges?........................171.6Notes......................................182AnalyticalMethods292.1Matchedasymptotics.............................302.1.1Asimpleadvection-di?usionequation................302.1.2Boundarylayersinincompressible?ows...............322.1.3Structureandd
5、ynamicsofshocks..................34iiiCONTENTS2.1.4TransitionlayersintheAllen-Cahnequation............362.2TheWKBmethod..............................392.3Averagingmethods..............................412.3.1Oscillatoryproblems.........................412.3.2Stochasticordinarydi?erential
6、equations..............452.3.3Stochasticsimulationalgorithms...................502.4Multiscaleexpansions.............................572.4.1Removingsecularterms.......................582.4.2Homogenizationofellipticequations................592.4.3HomogenizationoftheHamilton-Jacobiequatio
7、ns.........642.4.4Flowinporousmedia.........................672.5Scalingandself-similarsolutions.......................682.5.1Dimensionalanalysis.........................682.5.2Self-similarsolutionsofPDEs....................702.6Renormalizationgroupanalysis.....................
