Real-Space and Multigrid Methods

《Real-Space and Multigrid Methods》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、CHAPTER5Real-SpaceandMultigridMethodsinComputationalChemistryThomasL.BeckDepartmentsofChemistryandPhysics,UniversityofCincinnati,Cincinnati,OhioINTRODUCTIONReal-spacemethodsareiterativenumericaltechniquesforsolvingpartialdifferentialequationsongridsincoordinatespace.Thephysicalresponsesduetomanychem

2、icalphenomenaarerestrictedtodomainsthatarerelativelylocalinspace,andreal-spacemethodsarewellsuitedtoexploitthatphysicallocality.Inreal-spacemethods,theiterativeupdatesofthedesiredfunctionsrequireinformationonlyinasmallneighborhoodneartheupdatedpoint.Thedraw-backwiththisapproachisthat,iftheiterations

3、areperformedonlyonasingle(finest)grid,thesolvertendstostallduetothelong-wavelengthcomponentsoftheerrors.Multigridmethodsovercomethisstallingbyutilizinginformationfromawiderangeoflengthscales.Withtheincorporationofmultiscaleideas,solverscanoftenbedesignedforwhichthecostscaleslinearlywithsystemsize.Int

4、helast15years,real-spaceandmultigridmethodshavebeendevelopedforawiderangeofproblemsincomputationalchemistry.Thoseproblemsincludeelectronicstructure,PoissonandPoisson–Boltzmannequations,andtransportmodels.Thisreviewwillfirstgiveatutorialintroductiontoreal-spaceandmulti-gridmethodsandthenpresentsomecas

5、estudiesillustratinghowthesetechni-queshavebeenappliedinchemistry,nanomaterials,andbiology.ReviewsinComputationalChemistry,Volume26editedbyKennyB.LipkowitzandThomasR.CundariCopyright#2009JohnWiley&Sons,Inc.223224Real-SpaceandMultigridMethodsinComputationalChemistryPhysicalSystems:WhyDoWeNeedMultisca

6、leMethods?Moderncomputationalchemistryresearchfrequentlyexaminesproblemsthatrequiretreatingmultiple-lengthandtimescales.Itisinstructivetoconsiderfirsttwoexamplesystemsthatexemplifytheneedtodevelopmethodsforhand-lingmultiplescales.Thesetwoexamplescomefromrecentresearchprojectsintheauthor’sgroup;theyar

7、elimitedinscopebutrepresentativeofchallengingcomputationalproblemsinbiophysicsandnanoscience.Considerfirstionchannelsandtransporters,whicharelargeproteins1–3embeddedinbiologicalmembranes.Theirfunctioni