简介：BasedontheTaylorseriesmethodandLi’sspatialdifferentialmethod,ahigh-orderhybridTaylor–Lischemeisproposed.Theresultsofalinearadvectionequationindicatethat,usingtheinitialvaluesofthesquare-wavetype,aresultwiththirdorderaccuracyoccurs.However,usinginitialvaluesassociatedwiththeGaussianfunctiontype,aresultwithveryhighprecisionappears.Thestudydemonstratesthat,whentheorderofthetimeintegralismorethanthree,thecorrespondingoptimalspatialdifferenceordercouldbehigherthansix.Theresultsindicatethatthereasonforwhythereisnoimprovementrelatedtoanorderofspatialdifferenceabovesixistheuseofatimeintegralschemethatisnothighenough.TheauthoralsoproposesarecursivedifferentialmethodtoimprovetheTaylor–Lischeme’scomputationspeed.Amorerapidandhighprecisionprogramthandirectcomputationofthehigh-orderspacedifferentialitemisemployed,andthecomputationspeedisdramaticallyboosted.Basedonamultiple-precisionlibrary,theultrahigh-orderTaylor–LischemecanbeusedtosolvetheadvectionequationandBurgers’equation.