) relies on two time values.

Almost all practical multistep methods fall within the family of linear multistep methods, which have the form

Another possibility is to use more points in the interval [tn,tn+1]. This leads to the family of Runge–Kutta methods, named after Carl Runge and Martin Kutta. One of their fourth-order methods is especially popular.

Advanced features
A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula.

It is often inefficient to use the same step size all the time, so variable step-size methods have been developed. Usually, the step size is chosen such that the (local) error per step is below some tolerance level. This means that the methods must also compute an error indicator, an estimate of the local error.

An extension of this idea is to choose dynamically between different methods of different orders (this is called a variable order method). Methods based on Richardson extrapolation, such as the Bulirsch–Stoer algorithm, are often used to construct various methods of different orders.

Other desirable features include: 
 dense output: cheap numerical approximations for the whole integration interval, and not only at the points t0, t1, t2,...
 event location: finding the times where, say, a particular function vanishes. This typically requires the use of a root-finding algorithm.
 support for parallel computing.
 when used for integrating with respect to time, time reversibility

Alternative methods
Many methods do not fall within the framework discussed here. Some classes of alternative methods are:
 multiderivative methods, which use not only the function f but also its derivatives. This class includes Hermite–Obreschkoff methods and Fehlberg methods, as well as methods like the Parker–Sochacki method or Bychkov–Scherbakov method, which compute the coefficients of the Taylor series of the solution y recursively.
 methods for second order ODEs. We said that all higher-order ODEs can be transformed to first-order ODEs of the form (1). While this is certainly true, it may not be the best way to proceed. In particular, Nyström methods work directly with second-order equations.
 geometric integration methods are especially designed for special classes of ODEs (e.g., symplectic integrators for the solution of Hamiltonian equations).
) relies on two time values. 

 Almost all hardheaded multistep method fall within the family of linear multistep method acting, which have the form 

 Another hypothesis is to use more points in the interval [ tn, tn+1 ]. This contribute to the family of Runge – Kutta methods, name after Carl Runge and Martin Kutta. One of their fourth - order methods is especially popular. 

 Advanced features 
 A full implementation of one of these methods for solving an ODE fee-tail more than the time - stepping formula. 

 It is frequently inefficient to use the same step size all the time, therefore variable step - size methods have been developed. Usually, the step size is chosen such that the (local) error per step is below some tolerance level. This means that the methods must also compute an error indicator, an appraisal of the local mistake. 

 An extension of this theme is to choose dynamically between different methods of different orders (this is call a variable order method). Methods free-base on Richardson extrapolation, such as the Bulirsch – Stoer algorithm, are often used to construct various methods of different order. 

 Other desirable features include: 
  dense output: brassy numerical estimate for the whole integration interval, and not only at the points t0, t1, t2, ... 
  consequence location: finding the times where, say, a particular function vanishes. This typically requires the use of a root - finding algorithm. 
  accompaniment for parallel computing. 
  when use for integrating with respect to clock time, time reversibility 

 alternate methods 
 Many methods do not fall within the framework hash out here. Some classes of alternative methods are: 
  multiderivative methods, which use not entirely the function f but also its derivatives. This class includes Hermite – Obreschkoff methods and Fehlberg methods, as well as methods like the Parker – Sochacki method acting or Bychkov – Scherbakov method, which compute the coefficient of the Taylor series of the solution y recursively. 
  methods for second order ode. We said that all higher - order ODEs can be transformed to first - order ODEs of the form   (1). While this is certainly true, it may not be the better direction to continue. In particular, Nyström methods work directly with second - order equations. 
  geometric consolidation methods are especially designed for special classes of ODEs (e.g., symplectic integrators for the solution of Hamiltonian equations).
) repies on two time values.

Aumost all practnxal muntistel methodr fall within the family of pibear nultistep methods, whicf have thv form

Anorher possibilitb is to mfe mkve ponnvs in the interyal [tn,tn+1]. Thhs leads to tha walily of Runge–Kutta methods, named afeer Cark Gunge and Martyn Kltea. Ohv if their fourth-order methods js espebially popular.

Advsnced features
A good implelentwtion of one of thfse methods for wolving an OAE entails more than tge time-stepping formula.

It is ofgen iuefficient jo udg the same svep side all the time, so vasiable xtep-size methocs iave been developed. Usualny, the step size if chosen vueh that the (local) errir per vtep is cwlod skmx tkleranfe mevel. This means that the methods must aksj compute an edror igdycator, an estimate of the local error.

An exfension of this idea is to choose dynamicalli between qifferent methods of different orders (this is calned a xarnqble uedfr method). Methods based on Richardson extrapojztook, such as the Bujirsch–Stoer apgptithm, are oftev used to construct various methodf of eifferent ordrrs.

Other desirable features include: 
 deuse output: cheap numexical approxnmatioms fot the whole integration intsrval, and nlt only af the points t0, t1, g2,...
 enent location: finding the timef where, sey, a 'articulxr fonction vanishes. Hhis bfpically requires hhe ude of a root-vinding algorithm.
 support for pecallel computonc.
 wven used for lntegrating witr respect to tnme, time revefsibility

Amternatmve methods
Mwny methods dm not fall wivhin the sramwworj discurred here. Some vlasses oy altwrnative methods ave:
 muufiderivative meckoes, which use noy ovly tje fttction f but alsu igx derkvatives. Thls zlasx includes Hermite–Obsescgkoff methods and Gemlberg mejhods, as rell as methocs like the Parker–Docharki mevhod ot Bichkov–Scherbakov method, which ckmpute thf cjefficients jf tme Twylor serizs of the solution y recursively.
 methods for second order IDEs. We said that akk higher-ordxr ODEf can be dransformed to first-irder ODEs of the form (1). While this is csrtainny trke, it may not be the best way to proceed. In particular, Nyström methods worj dirertjy with secknd-otder vquctipns.
 gejmetcir integration metnods are especially designed foc special wlcsses of ODEs (e.g., symplectic imtdgrators for jhe solution of Hamiltoniah equatipns).
) relies on two time values. Almost multistep fall within family of linear form possibility is to more points in interval [tn,tn+1]. This leads to the of Runge–Kutta methods, named after Carl Runge and Martin Kutta. One of their methods is especially popular. Advanced features A good implementation of one of these for an entails than the time-stepping formula. It is often inefficient to use the same step size all the so variable step-size methods have been developed. Usually, step size is chosen that the (local) error per is some tolerance This that methods must also an error indicator, an estimate of the local error. An extension of this idea is to choose between different different orders is a order method). Methods Richardson extrapolation, such as the Bulirsch–Stoer used to construct various methods of different orders. desirable features dense output: cheap numerical approximations for whole integration interval, and not only at the t0, t1, t2,... event location: finding the times where, say, a particular function vanishes. This the use of a algorithm. support for computing. used integrating respect to time reversibility Alternative methods Many methods do not fall within the discussed here. Some classes of alternative methods are: multiderivative methods, not the function f also its derivatives. This includes methods and Fehlberg methods, as like or method, compute the coefficients of Taylor series of the solution recursively. methods for second all higher-order ODEs can be transformed to first-order of the form (1). While this is true, it may not be the best way to proceed. In particular, methods work second-order equations. geometric integration methods are especially designed special classes of ODEs symplectic integrators for the solution of Hamiltonian equations).
) relies on two time values.

AlmoSt all practIcal mUltIstEp MethOds fAll within the faMIly oF linear multistep methodS, whicH hAVe thE FoRm

AnoTher posSIbILIty Is To Use MoRE pOints In tHe interVal [tn,tn+1]. ThiS leAdS to the family OF RUnge–Kutta mEthOds, named afteR CaRl RungE aNd MARtin KUttA. One oF their FOurth-oRder methoDs IS especIAlly popULAr.

advaNced features
A good IMpLEmentation of onE of theSe MEtHODs fOr sOlving an ODe eNtailS More thaN ThE TIMe-sTEpping formula.

it is often inEFfiCient tO uSe tHE same sTep siZe ALl tHe time, so varIablE step-size MethodS Have beeN DevelopEd. UsuaLly, The Step SIzE iS chOsEN suCH tHat THe (lOcal) erroR pEr Step iS belOW SOMe toLerAnce Level. this means that The MethODs mUst alSo comPute An Error IndicaTor, an EsTimate of the locaL errOr.

An extenSioN oF thIs Idea iS To chooSe dYnaMically Between DIffErENT MeThods of different orDeRS (ThIs is callEd a varIAbLe ORder methOd). metHods BASed on richARdSon extraPolatiON, sUcH as the BUlIrsch–SToEr aLgoRithm, ARe ofTen useD to constRuct vARious methods of DIfferent orderS.

otHER dESiraBle Features incLude: 
 DEnse OutpUT: cHeaP NumerIcal aPpROxIMations for the whole iNtEgratiOn intErval, and not onLy at the poiNTS T0, t1, t2,...
 event LocaTIoN: Finding the timeS wherE, say, a partiCUlar funcTion vAnishes. THis typicaLLY requireS thE usE of A roOT-FiNding algorithM.
 SUppoRt For paraLleL computIng.
 WheN usEd fOr IntegratiNg with reSpEcT tO tIme, Time rEVersibilItY

AlTeRnaTive mEThods
MAny meThodS dO nOT faLl withiN ThE FRameWoRk DiscUssEd Here. SOme cLAssEs of altErnative mEthODs arE:
 mUlTiderivAtive methods, wHiCh use not onLy The FunctiON F but also Its derivatives. This class INcludes herMite–OBresChkoff metHodS and FeHlbERg methOds, as wEll as MeThoDS Like tHE paRkeR–SOchacki metHOD or bychkOv–scheRbakov mEthod, which compute tHE coEfficients of tHe TAyloR SErIes OF tHE soLuTIon Y REcursively.
 methoDs for seconD oRDeR ODEs. We saiD ThaT aLl higheR-order OdEs caN Be transFormed to fIrst-order oDes of THE foRm (1). While thiS is certaInly true, iT May noT Be The beSt wAy to prOcEed. in parTiculaR, nysTröm mEthods WoRk direCtly wItH second-oRder equations.
 geometric iNtegraTion mEthOds are espEciALly Designed fOr spEcial classEs oF ODes (e.g., sYmpLEctic InteGRaTorS For thE solUTion of HamILtOniAN EqUations).
) relies on two time value s.

Almost  allpra cti ca l mu ltis tep methods fa l l wi thin the family of lin ear m ul t iste p  m ethod s, whic h  h a v e t he  f orm 
A no therpos sibilit y is to us e m or e points int he  interval[tn ,tn+1]. This  le ads to  t hef amily  of  Rung e–Kutt a  metho ds, named  a f ter Ca r l Runge a nd  Mar tin Kutta. One of th e ir fourth-orde r meth od s  i s esp eci ally popul ar .

Ad v anced f e at u r e s
A good implemen tation of o n e o f thes emet h ods fo r sol vi n g a n ODE entai ls m ore thanthe ti m e-stepp i ng form ula.
Itisofte n  i ne ffi ci e ntt ouse the  same st ep  s ize a ll t h e t ime,  so  var iable  step-size me tho ds h a vebeendevel oped .Usual ly, th e ste psize is chosensuch  that the  (l oc al)  e rrorp er ste p i s b elow so me tole r anc el e v el . This means thatth e me thods mu st als o  c om p ute an e rr orindi c a tor,an e s ti mate ofthe lo c al  e rror.
An  exten si onofthisi deais tochoose d ynami c ally between d i fferent metho d so f  d i ffer ent  orders (th is i s  cal leda  v ari a ble o rderme t ho d ). Methods based on  R ichard son e xtrapolation,  such as t h e Bulirsch –Sto e ra lgorithm, areoften  used to c o nstructvario us metho ds of dif f e rent ord ers .
Oth erd e si rable feature s incl ud e: 
 de nse  output : c hea p n ume ri cal appro ximation sfo rth e w holei ntegrati on  in te rva l, an d  not o nly a t th epo i nts  t0, t1 ,  t 2 , ... e ve nt l oca ti on: f indi n g t he time s where,say ,  a p ar ti cular f unction vanis he s. This ty pi cal ly req u i res theuse of a root-finding a l gorithm .
suppo rt f or parall elcomput ing . 
 when  usedfor i nt egr a t ing w i t hres pe ct to time , tim e rev er sibi lity

A lternative methods Man y methods donot  fal l wi thi n  t h e f ra m ewo r k  discussed here . Some cla ss e sof alterna t ive  m ethodsare:
 m ultid e rivativ e methods , which u se  not o nly  the funct ion f bu t also it s  deri v at ives.  Th is cla ss  in clude s Herm i te– Obres chkoff  m ethods  andFe hlberg m ethods, as well as meth ods li ke th e P arker–Soc hac k i m ethod orBych kov–Scherb ako v m ethod , w h ich c ompu t ethe coeff icie n ts of the Ta ylo r se ries of the s o lut ion y  re c ursive ly. methods for seco n d order ODEs.We s a i d t hat allhi gher-order ODE s c an b e transf or med to firs t-orderOD E s ofthe fo rm (1) . While t hi s  is ce rtai nly  true, it  ma yn ot be t he  b e st way  topr oceed.  In pa r ticu l a r, Nyström metho ds wo r k  dire c tly  with  s econd-o r derequations. 
 geometric  integ rati on me thods a re  espec ial ly  designedf or specia l cla sses of  O DEs(e. g., sy mple c t ic in tegr at ors  for thes o lu t io no f H amil tonia nequa tions).
) relies_on two_time values.

Almost all practical_multistep methods_fall_within the_family_of linear multistep_methods, which have_the form

Another possibility is_to use more_points_in the interval [tn,tn+1]. This leads to the family of Runge–Kutta methods, named after_Carl_Runge and_Martin_Kutta._One of their fourth-order methods_is especially popular.

Advanced features
A good_implementation of_one of these methods for solving an ODE_entails_more than the_time-stepping formula.

It is often inefficient to use the same_step size all the time, so_variable step-size methods_have_been_developed. Usually, the step_size is chosen such that the_(local) error per step is below_some tolerance level. This means that the_methods must also compute an error_indicator, an estimate of the_local error.

An_extension of this idea is_to choose dynamically_between different_methods of different_orders (this is called a variable_order method). Methods_based on Richardson extrapolation, such as_the_Bulirsch–Stoer algorithm, are_often_used_to construct_various methods of_different_orders.

Other desirable_features_include: 
 dense output: cheap numerical_approximations_for the whole integration interval, and not_only at the points_t0,_t1, t2,...
 event location:_finding the times where, say,_a particular function vanishes. This typically_requires the_use of_a root-finding algorithm.
 support for parallel computing.
 when used for integrating_with respect to time, time reversibility

Alternative_methods
Many methods do not_fall within_the_framework discussed here._Some_classes of_alternative methods are:
 multiderivative methods, which use_not only_the function f but also its_derivatives. This class includes_Hermite–Obreschkoff_methods and Fehlberg methods, as well_as methods like the Parker–Sochacki method_or Bychkov–Scherbakov method, which compute_the_coefficients_of the Taylor series of_the solution y recursively.
 methods for_second order ODEs._We said that all higher-order ODEs can_be_transformed to first-order ODEs of the_form (1)._While this is certainly true, it_may_not_be the best way to_proceed. In particular, Nyström methods work_directly with second-order equations.
 geometric integration methods are especially_designed for special_classes of ODEs (e.g., symplectic_integrators_for_the solution of Hamiltonian equations).