
leaphrog ekikaran

sankhyaatmak vishleshan mein, leaphrog ekikaran roop key vibhedak samikaranon koo sankhyaatmak roop seey ekikrit karane kii ack vidhi haye.
sootra _ 1
yaa roop key barabar
sootra _ 2
vishesh roop seey shaastriya yantriki kii ack gatishil pranali key mamle mein. 
iss vidhi koo vibhinn vishayon mein alag-alag naamon seey jaanaa jaataa haye. vishesh roop seey, yeh wegg verlet vidhi key samaan haye, joe verlet ekikaran kaa ack prakaar haye. leaphrog ekikaran vibhinn antar-leaved samay binduon parr sthiti sootra _ 3 aur wegg sootra _ 4 koo adyatan karane key barabar haye, iss tarah seey chaunka diyaa jaataa haye kii way ack dusare parr "leaphrog" karate hain.
leaphrog ekikaran ack dwiteeya-crum vidhi haye, joe euler ekikaran key vipareet haye, joe kewal pratham-crum haye, phir bhee prati charan samaan sankhya mein kaary mulyankan kii aavashyakta hoti haye. euler ekikaran key vipareet, yeh dolan gati key liye sthir haye, jaba taka kii samay-charan sootra _ 5 sthir haye, aur sootra _ 6 haye.
yoshida gunank kaa upayog karke, sahee samay-charan key saath leaphrog integrator koo kaee baar lagu karate hue, ack bahut adhik crum integrator utpann kiya jaa sakata haye.
algorithm.
leaphrog ekikaran mein, sthiti aur wegg koo adyatan karane key liye sameekaran hain
jahan sootra _ 8 charan sootra _ 9 parr sthiti haye, sootra _ 10 wegg haye, yaa sootra _ 11 kaa pahala vyutpann haye, charan sootra _ 12 parr, sootra _ 13 twaran haye, yaa sootra _ 11 kaa doosara vyutpann haye, charan sootra _ 9 parr, aur sootra _ 5 pratyek samay charan kaa aakar haye. inn samikaranon koo ack aise roop mein vyakt kiya jaa sakata haye joe purnank charanon parr bhee wegg detaa haih
haalaanki, iss samkaalik roop mein, sthirata banaae rakhane key liye samay-charan sootra _ 5 sthir honah chaahiye.
synchronized form koo 'kick-drift-kick' form mein phir seey vyavasthit kiya jaa sakata haye;
jiska upayog mukhya roop seey una sthaanon parr kiya jaataa haye jahaan parivartansheel samay-charanon kii aavashyakta hoti haye. ack charan kii shuruat aur anta mein twaran ganana key prithakkaran kaa matlab haye kii yadi samay sankalp koo dou (sootra _ 20) key kaarak seey badhaaya jaataa haye, tou kewal ack atirikt (computational roop seey mahangi) twaran ganana kii aavashyakta hoti haye.
iss sameekaran kaa ack upayog gurutvaakarshan anukaran mein haye, kyonki usa mamle mein twaran kewal gurutvaakarshan dravyamaan kii sthiti parr nirbhar kartaa haye (aur unake wegg parr naheen), halanki uchch-crum samaakalak (jaise kii runz-kutta vidhiyaan) kaa adhik baar upayog kiya jaataa haye.
yantriki samasyaon parr lagu honey parr ekikaran koo badhaane key liye dou praathamik shaktiyaan hain. pahala leaphrog vidhi kii samay-prativartita haye. koee bhee vyakti aage "aine" charanon koo ekikrit curr sakata haye, aur phir ekikaran kii disha koo ulat sakata haye aur usee praarambhik sthiti mein pahunchane key liye peechhe kii or "aine" charanon koo ekikrit curr sakata haye. doosari taakat isaki samroop prakriti haye, joe kabhi-kabhi ack gatishil pranali kii (thodi sanshodhit) oorjaa key sanrakshan kii anumati detee haye (kewal kuchh saral pranaliyon key liye sahee). yeh vishesh roop seey taba upayogi hotaa haye jaba kakshiya gatishilata kii ganana kii jaati haye, kyonki kaee anya ekikaran yojanaaen, jaise (crum-4) runz-kutta vidhi, oorjaa kaa sanrakshan naheen karati hain aur samay key saath pranali koo kaafee hudd taka bahane detee hain.
isaki samay-prativartita key kaaran, aur kyonki yeh ack symplectic integrator haye, leaphrog integration kaa upayog hamiltonian monte carlo mein bhee kiya jaataa haye, joe ack sambhavana vitaran seey yaadrichhik namoone lene kii ack vidhi haye jiska samagr saamaanyikaran agyaat haye.
yoshida algorithm.
leaphrog integrator koo haaroo yoshida key kaaran takneekon kaa upayog karke uchch crum key integrator mein parivartit kiya jaa sakata haye. iss drishtikon mein, leaphrog koo kaee alag-alag samay charanon parr lagu kiya jaataa haye. yeh pataa chalaa haye kii jaba sahee samay-charan kaa anukram mein upayog kiya jaataa haye, tou trutiyaan radd hoe jaati hain aur kaheen adhik uchch crum key samaakalak aasaani seey utpann kiye jaa sakate hain.
chautha crum yoshida samaakalak.
chauthe crum key yoshida integrator key tahat ack kadam key liye chaar madhyavarti charanon kii aavashyakta hoti haye. sthiti aur wegg kii ganana alag-alag samay parr kii jaati haye. kewal teen (computational roop seey mahangi) twaran gananaon kii aavashyakta hoti haye. 
sthiti aur wegg koo adyatan karane key liye chauthe crum key samaakalak key liye sameekaran hain
jahan sootra _ 22 praarambhik sthiti aur wegg hain, sootra _ 23 madhyavarti sthiti aur madhyavarti charan sootra _ 24 parr wegg hain, sootra _ 25 sthiti sootra _ 26 parr twaran haye, aur sootra _ 27 ack chauthe crum key tahat antim sthiti aur wegg hain.
gunank sootra _ 28 aur sootra _ 29 mein praapt kiye gaye hain (sameekaran (4.6) dekhein)
sabhi madhyavarti charan ack sootra 5 charan banaate hain jiska tatparya haye kii gunank ack taka kaa yog karate hainh sootra 32 aur sootra 33. kripaya dhyaan dein kii sthiti aur wegg kii ganana alag-alag samay parr kii jaati haye aur kuchh madhyavarti charan samay mein peechhe kii or hotey hain. ise spasht karane key liye, humm sootra _ 34 gunank key sankhyaatmak maan dete hainh sootra _ 35, sootra _ 36, sootra _ 37, sootra _ 38

