二階常微分方程(組)初值問題的Legendre-Gauss和廣義Laguerre-Gauss配置方法.pdf

1、上海大學(xué)博士學(xué)位論文二階常微分方程（組）初值問題的LegendreGauss和廣義LaguerreGauss配置方法姓名：嚴(yán)建平申請學(xué)位級別：博士專業(yè)：計算數(shù)學(xué)指導(dǎo)教師：郭本瑜20080705ABSTRACTThenumericalmethodsforsolvingthedifferentialequationsisoneofthesubjectsofcomputationalmathematics笛wellasanimportantp

2、artoflargescalescientificcomputingInthisthesis，thespectralmethod，whichisoneofthethreemainnumericalmethodsfordifferentialequations，isinvestigatedThefascinatingmeritofspectralmethodisitshighaccuracyie，thesmoothertheexactso

3、lution，themoreaccuratethenumericalresultsTherefore，ithasbeenappliedsuccessfullytonumericalsimulationsofmanypractialproblems，suchasthenumericalsimulationofquantummechanics，weatherprediction，oceanproblems，andsoonManypracti

4、calproblemsarisinginscienceandengineeringaregovernedbyini—tialvalueproblems(IVPs)ofordinarydifferentialequations(ODEs)TherehavebeenfruitfulresultsonnumericalmethodsforfirstorderODEs，see，eg，Butcherp8】，Hairer，NorsettandWan

5、ner【46】，HairerandWanner【47】，Higham【49】，Lambert【5Zl，Stetter(so]andStuartandHumphries[s2]ForHamiltoniansystems，werefertothesymplecticdifferencemethod，seeFeng【26]，F(xiàn)engandQin【25】，Hairer，LubichandWanner[45】，andSanzSernaandCal

6、vo[71]Iti8alsoimportantandinterestingtoconsiderIVPsofsecondorderODEsForinstance，afterspacialdiscretization，manynonlinearwaveequations，suchasKleinGordenandsineGordenequations，arereducedtocertainsystemsofsecondorderODEsWem

7、ayreformulatesuchproblemstosomesystemsoffirstorderODEsandthensolvethemnumericallyWhereas，forsavingwork，itseemsreasonabletosolvethemdirectlysometimes，see，eg，F(xiàn)ehlberg【221，Kramarz【55】，KuoandVazquez【56]，F(xiàn)ranco[261，VigoAguiar

8、andR舡nosIS41，andKonguetofandSimos【541However，SOfar，thereisnonumericalmethodwiththespectralaccuracyforinitialvalueproblemsofsecondorderODEsThemaindifficultyishowtodesignproperalgorithmsandanalyzetheirnumericalerrorsprecis

9、elyInthisthesis，weproposedtwonumericalintegrationalgorithmsfortheIVPsofsecondorderODEsNumericalsimulationsindicatetheirreliableefficiencyandcoincidewellwiththeoreticalanalysisInthefirstchapter，weoutlinethemotivationandth

10、emainresultsInthesecondchapter，westudyanewnumericalmethodforIVPsofsecondorderODEsWeproposedanewcollocationmethodbasedonLegendreGaussinterpolation，andproveditsspectralaccuracyNumericalresultsindicatethemeritsofthisapproac