SS 65 Spng Quae 5 Knec Sulaon Lecue une 9 5
An aple of an lecoagnec Pacle Code A an eaple of a knec ulaon we wll ue a one denonal elecoagnec ulaon code called KMPO deeloped b Yohhau Oua and Hoh Mauoo. Moe dealed nfoaon ncludng nucon on how o un he code aalable fo hp://www.eapub.co.jp/e-lba/cpp/nde.hl. We hae ncluded a eon of he code on he cla web page. KMPO an eaple of cla of code known ae pacle n cell code. In hee code he euaon of oon of he pacle n a plaa oled elf-conenl. The pacle oon fo boh on and elecon ae calculaed n elecc and agnec feld. The eulng change n he elecc and agnec feld ae calculaed and hen bul back no he pacle calculaon. The KMPO code uffcenl fleble ha can be ued fo boh e pacle calculaon (.e. no elf-conen feld) o a an elecoac code (change n he elecc feld onl).
The ac Poble The elecoagnec feld' ae gen b Mawell' uaon whee ρ c ε and µ ae he cuen den chage den lgh peed elecc pe and agnec peeabl epecel. µ c ρ ε
The ac Poble Fo h eaple he gd aued o be one-denonal. The elecc feld X u af Poon' euaon nall. Noe h onl ha o be done a he begnnng of he ulaon nce Apee' Law wll aue oled auoacall. The agnec feld coponen hould af In a one-denonal e h ake conan. The code ole Mawell' euaon fo he elecc feld ( ) and he agnec feld ( ). The cuen den ( ) and he chage den ρ ae copued fo he oon of a lage nube of pacle. ρ ε Fo pacle of chage and a he euaon of oon d d ( ) d d In ulaon e and can be defned abal poded ε µ c
The Gd Saeg The elecc feld cuen den and agnec feld ae defned a paal gd pon whle he pacle can ake aba poon. Inepolaon ued o ge he feld fo calculang he ajecoe. Full nege gd ae a (3 N ) and half-nege gd a ( ½). The uane n he ulaon ae defned a hown on he gh. The elecc feld and he cuen den u be agned o he ae gd nce he cuen den conbue decl o he e dependen elecc feld. The e and paal deae n Apee' Law and Faada' Law ae ceneed dffeence. / 3/ ρ Ε ρ Ε
How he Schee Wok: Te Seuencng Thee ae wo e euence- full nege e (n) and half-nege e (n/). acall he elecc feld negaed a full nege e and he agnec feld a half nege e. The negaon done b ung he leap-fog ehod. The agnec feld adanced wce b a half ep o oban needae alue fo he pacle puhng feld a he full-nege e. 3 5 4 6 7 Pacle poon a he full nege e and eloce a he half nege e ae adanced b he leap-fog ehod. The poon ae adanced wce o he cuen den can be calculaed a he e nege e.
Dgeon on he Leap-Fog Mehod A Leap-Fog ehod econd ode accuae n e a well a pace. Conde agan he adecon euaon we ued a an eaple when dcung he oluon of he MHD euaon. u The oluon ake on he followng fo n a Leap-Fog e. The e leel (he n') n he e deae e "leapfog" oe he e leel n he pace deae e. Noe he u n- and u n u be oed n ode o calculae u n. The on Neuann abl anal ge o he Couan condon eued. Thee no aplude dpaon. u ( n n u u ) n n u j u j j j ξ ξ n k
The Couan Condon Aue a uan ha a ucue wh a waenube k a feuenc ω. The deae becoe A ( k ω ) A e ( ) A( ) A A ( k ) ( k ) e e n( k ) A A A ( ) ( ) Reebe ha epeen o ha n he dffeence euaon he waenube k eplaced b K. n K ( k ) A
The Couan Condon Slal he feuenc ω eplaced b Ω whee n Ω ( ω ) The dpeon elaon fo elecoagnec wae a acuu ω c k The nuecal dpeon elaon becoe Ω c K Fo he au waennube occu fo n ( ω ) ka π If c > ω becoe cople and we hae a nuecal nabl. c If c he e agnall able. The he Couan condon.
The Debe Lengh To aod nabl caued b he gd pacng hould be cloe o he Debe lengh λ e gen b h e whee he and ω pe ae he heal eloc and he plaa feuenc of elecon. In he acual KMPO code he gd pacng e o λ e ω pe 3λ e
How he Code Wok: Inalaon Defne he chage den. A he gd pon X calculaed b whee he pacle hape funcon gen b The nal elecc feld calculaed fo Poon' euaon b N p j j j X W ρ W > ε ρ
How he Code Wok: Solng Mawell' uaon The cuen den b ung chage coneaon and ae calculaed b Th ge a half-nege gd bu we need a whole nege gd o he aeage ued Once we know cuen den we can calculae and b ung Mawell' euaon. ρ ρ j N j j X W p
How he Code Wok: Solng Mawell' uaon Adancng he elecc feld Adancng he agnec feld c c
How he Code Wok: Solng he uaon of Moon d d d d ) ( Ne we u ole he euaon of oon The dffeence euaon Defne
How he Code Wok: Solng he uaon of Moon The acual code ue he unean-o ehod h fou ep ehod aue c coneaon of knec eneg n he ccloon oon. 4 / 3
How he Code Wok: Solng he uaon of Moon 3 In adancng he eloc fo -/ o / we need he elecc and agnec feld a he e a he pacle poon (). The feld ae lneal nepolaed fo adjacen gd pon. A pacle hould no be nfluenced b he feld due o own chage. The elecoac feld defned on half nege gd bu ha o be elocaed o full-nege gd befoe he nepolaon o cancel he elecoac elffoce. Slal he agnec feld ha o be elocaed befoe nepolaon fo full-nege gd o half nege gd o cancel he agneoac foce due o he cuen den The effec of h goe awa f µ
How he Code Wok: Solng he uaon of Moon 4 In one e ep he pacle poon adance wce each b a half e ep.