2N-Storage Low Dissipation and Dispersion Runge-Kutta Schemes for Computational Acoustics

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1、JOURNALOFCOMPUTATIONALPHYSICS143,674–681(1998)ARTICLENO.CP985986NOTE2N-StorageLowDissipationandDispersionRunge-KuttaSchemesforComputationalAcoustics1.INTRODUCTIONForphysicalproblemsthatinvolveaccuratetime-dependentwavepropagation,asthosearisinginacoustics,theusualrequirementofahigh-ordertrun

2、cationerrordoesnotguaranteethatanumericalmethodyieldsaccurateresults.Indeed,ashasbeenpointedoutmainlyin[1],thedissipationanddispersionpropertiesofthenumericalmethodareveryimportantforcomputingwavesolutionsofsystemsofpartialdifferentialequations.Thisisvalidforboththespatialandthetimediscretiz

3、ationmethods.TheexplicitRunge-Kutta(RK)methodsarewidelyusedtodiscretizethetimederivativebecauseoftheiradvantagesthatincludeflexibility,largestabilitylimits,andeaseofprogramming.Huandco-workers[2]showedthatthedissipationanddispersionpropertiesoftheRKmethodsdependontheircoefficientsandoptimizedt

4、hemfortheconvectivewaveequation,obtainingwhattheycalledlow-dissipationanddispersionRunge-Kutta(LDDRK)methods.Thesemethodsaremoreefficientthanclassicalones,intermsofworkrequiredforagivenaccuracy,forwavepropagationproblems.Forlargesizephysicalproblems,memoryrequirementsmaybecomeexhaustive.Theyc

5、anbedecreasedusingspecialRKschemesthatcanbewrittensuchthatonly2N-storageisrequired,whereNisthenumberofdegreesoffreedomofthesystem(i.e.,numberofgridpoints£numberofvariables).TodesignsuchRKschemes,enoughfreecoefficientsmustexistsuchthatadditionalconditionsholdbetweenthem.Williamson[3]firstshowed

6、thatallsecond-orderandsomethird-ordermethodscanbewrittenin2N-storageform.Healsoshowedthatfourth-orderfour-stagemethodscannotbewritteninthisway.Byallowingadditionalstagesandusingtheresultingnewfreecoefficientstoimposethe2N-storageconstraints,CarpenterandKennedy[4]devisedafourth-order,five-stage

7、sRKmethodthatiscompatiblewiththeclassicalfourth-ordermethodwhichhoweverrequiresatleast3Nstorage.Huetal.[2]provide3N-storageimplementationsoftheLDDRKschemes.Thesearevalidforlinearproblemsonly,inthesensethattheyturntosecondorderaccuracywhenappliedton