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1、Finitevolumemethods:foundationandanalysisTimothyBarth1andMarioOhlberger21NASAAmesResearchCenter,InformationSciencesDirectorate,MoffettField,California,94035,USA2InstituteofAppliedMathematics,FreiburgUniversity,Hermann-Herder-Str.10,79104Freiburg,GermanyandCSCAMM,UniversityofMaryland,4127CSI
2、CBuilding,CollegePark,Maryland,20742-3289,USAABSTRACTFinitevolumemethodsareaclassofdiscretizationschemesthathaveprovenhighlysuccessfulinapproximatingthesolutionofawidevarietyofconservationlawsystems.Theyareextensivelyusedinfluidmechanics,meteorology,electromagnetics,semi-conductordevicesimu
3、lation,modelsofbiologicalprocessesandmanyotherengineeringareasgovernedbyconservativesystemsthatcanbewritteninintegralcontrolvolumeform.Thisarticlereviewselementsofthefoundationandanalysisofmodernfinitevolumemethods.Theprimaryadvantagesofthesemethodsarenumericalrobustnessthroughtheobtentiono
4、fdiscretemaximum(minimum)principles,applicabilityonverygeneralunstructuredmeshes,andtheintrinsiclocalconservationpropertiesoftheresultingschemes.Throughoutthisarticle,specificattentionisgiventoscalarnonlinearhyperbolicconservationlawsandthedevelopmentofhighorderaccurateschemesfordiscretizin
5、gthem.Akeytoolinthedesignandanalysisoffinitevolumeschemessuitablefordiscontinuitycapturingisdiscretemaximumprincipleanalysis.Anumberofbuildingblocksusedinthedevelopmentofnumericalschemespossessinglocaldiscretemaximumprinciplesarereviewedinoneandseveralspacedimensions,e.g.monotonefluxes,TVDdi
6、scretization,positivecoefficientdiscretization,non-oscillatoryreconstruction,slopelimiters,etc.Whenavailable,theoreticalresultsconcerningaprioriandaposteriorierrorestimatesaregiven.Furtheradvancedtopicsarethenconsideredsuchashighordertimeintegration,discretizationofdiffusiontermsandtheextensi
7、ontosystemsofnonlinearconservationlaws.keywords:finitevolumemethods,conservationlaws,non-oscillatoryapproximation,discretemaximumprinciples,higherorderschemesContents1Introduction:Scalarnonlinearconservationlaws21.1Themethodofcharacteristics....................
