Numerical solution of differential equations

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1 Numecal soluon of ffeenal euaons Devng an solvng ffeenal euaons DE s a common ask n compuaonal eseac. Many pyscal laws/elaons ae fomulae n ems of DE. Mos connuum smulaon meos ae base on soluon of DE. Aloug e fne ffeence an e fne elemen meos ypcally use o solve DE ae ofen nuvely assocae w lage-scale macoscopc poblems, s assocaon s naeuae. Te meos ae no calbae o any pyscal me o leng scale. Numecal meos fo solvng DE s a vas an complex subjec aea. In s lecue we wll only scac e suface by befly scussng seveal meos use fo solvng a sysem of onay DE elevan o Pacle Dynamcs meos. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

Pacle Dynamcs Pacle Dynamcs moels: Pyscal sysem s epesene as a ses of pacles ae an enses fels evolvng ove me.

Examples: Moon of asonomcal boes une e acon of gavaonal foces Moon of aoms une e acon of neaomc foces n Molecula

yoynamcs, Pacle-n- Cell an oe uas-pacle meos as alenaves o connuum appoaces Dsspave Pacle Dynamcs smulaon of

2 Pacle Dynamcs Pacle Dynamcs moels: Pyscal sysem s epesene as a ses of pacles ae an enses fels evolvng ove me. MD smulaon of cack popagaon P. Vassa e al. Examples: Moon of asonomcal boes une e acon of gavaonal foces Moon of aoms une e acon of neaomc foces n Molecula Dynamcs MD meo Coase-gane escpon of e ynamcs of flus an polymec sysems by Dsspave Pacle Dynamcs meo Smooe Pacle yoynamcs, Pacle-n- Cell an oe uas-pacle meos as alenaves o connuum appoaces Dsspave Pacle Dynamcs smulaon of Rayleg-Taylo nsably Dzwnel e al. Molec. Smul., 97, 1999 Rayleg-Taylo nsably n aomsc smulaons S e al. J. Pys. Cem. C 11, 1659, 017 Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

3 Pacle Dynamcs Onay Dffeenal Euaons ODE In Pacle Dynamcs moels laws of moon ae ypcally onay ffeenal euaons onay means a unknown funcons epen only on one vaable, ypcally me. Conse sysem of N pacles. Tajecoy of a pacle of mass of m, une e acon of foce F can be escbe by e Newon s euaon of moon - e secon oe ffeenal euaons e oe of a DE s e ges oe of any of e evaves n e euaon: m v F N euaons:, m 1,..., x N, F x n D space x x y z, y, z, F F, F, F, m ese N secon oe ODE can be ewen as a sysem of 6N fs oe euaons: x x, v y x x y y y y z, z v, v F v, F, F y m z = 1,,,N Teefoe, n geneal, ou ask s o solve a sysem of euaons u/ = fu, w nal conons gven by u 0 = u 0 m m F z v z z F m Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

4 Fne ffeence meos fo Pacle Dynamcs poblem Tus, we nee o solve a sysem of fs oe ODEs. Analycal soluon s ffcul an ofen mpossble fo a sysem of moe an neacng pacles foce acng on a pacle epens on posons of all oe pacles an negaon of e euaon woul nvolve negang ove a sum. Teefoe we ave o use a numecal appoxmaon fo e negal ove me. Le s conse, fo smplcy only one euaon u/ = fu,, u 0 = u 0 u u 0 0 Ts ype of poblem s calle nal value poblem, n conas o bounay value poblem, n wc values of u ae gven a wo en pons. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

5 Fne ffeence meos fo Pacle Dynamcs poblem Tus, we wan o solve e followng fs oe ODE euaon u/ = fu,, u 0 = u 0 u u 0 0 Fne ffeence ecnue s base on wo appoxmaons: 1. Dscezaon of me: a g of meseps s nouce.. Replace e ffeenal euaon by e coesponng fne ffeence euaon: u u = u+ u u lm 0 u u u 0 Fne ffeence meo allows us, sang fom e nal value u 0, o calculae values of u 0 +n one sep a a me. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

6 Eule s Meo Inaccuae an can be unsable: soul no be use! Fo n = 0 + n, 0 n las 0 / an appoxmae soluon s gven by u n+1 = u n + *fu n, n u Poo guess Lnea appoxmaon s use o ge e nex pon u n Real value n n+1 n+ Te man poblem w Eule meo s a only nfomaon fom e begnnng of e neval s use o exapolae e value a e oe se of e neval. In oe wos, only slope of e funcon s aken no accoun, e cuvaue s gnoe. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

7 u n 1 u n u n Eo n Eule s meo Le s use Taylo expanson o ge e exac soluon: p p u u! n... p! n p... Assumng a u n s exac, e esmaon of local eo ue o e uncaon s u n 1 u n f u, O n Ts s a local eo. As accumulaes ove e -1 nevals e global eo s O. Tus, s s a fs-oe meo. Ta s, e uncaon eo s lnealy popoonal o e me neval. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

8 Mofe Eule s meo o secon-oe Runge-Kua meo Te accuacy of e appoxmaon can be mpove by evaluang e funcon f a wo pons somemes calle e mpon meo: k1 = * fu n, n k = * fu n +k1/, n +/ u n+1 = u n + k + O u u n k1 Bee guess Real value n n+1 n+ Ts bee exapolaon uses e fac a e aveage slope of e funcon s moe closely appoxmae by e slope a e cene of e neval k/. Te meo s neffcen an only moeaely accuae bu s use n calculaons somemes. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

9 Fou-oe classcal Runge-Kua meo k1 = * fu n, n u k1 k = * fu n +k1/, n +/ k = * fu n +k/, n +/ k = * fu n +k, n + u n+1 = u n + k1/6 + k/ + k/ + k/6 + O 5 u n n n+1 k k k 1. Use evave a e begnnng of e neval o ge fs al mpon. Use evave a e fs mpon o ge secon al mpon. Use evave a e secon mpon o ge a al enpon. Inegae by Smpson s ule, usng aveage of wo mpon esmaes Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

10 Fou-oe classcal Runge-Kua meo Te global uncaon eo of classcal Runge-Kua s O an s aceve by compung funcon f fou mes a eac mesep. As compae o e Eule s Meo, Te Runge-Kua oes mes as muc calculaons a eac mesep bu nees muc smalle numbe of seps fo e same me neval o aceve a gven accuacy. Ts s e mos popula meo fo many sysems nvolvng few egees of feeom. I aceves goo accuacy w moes compuaonal complexy. Numecal Recpes book: Fo many scenfc uses, fou-oe Runge-Kua s no jus e fs wo on ODE negaon, bu e las wo as well. In fac, you can ge pey fa on s ol wokose, especally f you combne w an aapve sepsze algom. Bu Runge-Kua s almos neve use n pacle ynamcs meos a nvolve many egees of feeom. Te man eason fo s vey g compuaonal cos. Te g-an se of e euaon as o be evaluae mes. Calculaon of funcon f foce s by fa e mos me consumng pa of any pacle meo. An aonal mno poblem s a e meo lacks e me-evesal symmey of e Newon s euaons. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

11 Specal meos fo pacle ynamcs Secon-oe ffeenal euaons n wc fs-oe evaves o no appea ae foun so feuenly n apple poblems, paculaly ose asng fom e law of moon, a specal meos ave been evse fo e soluon. Te ea s o go ecly fom e secon evaves o e funcon self wou avng o use e fs oe evaves. m F Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

12 Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle Vele algom fo F m + = F/m + O O 6 O 6 Le s we wo -oe Taylo expansons fo a + an - an sum em ogee O

13 Vele algom + = F/m + O Ts algom s one of e mos commonly use pacle ynamcs compuaonal meos. I s smple o mplemen, accuae an sable an eo mae a any gven sep ens o ecay ae an magnfy lae on. I s me evesble n pacce, accumulaon of compuaonal ounoff eos evenually beaks evesbly of e calculaon anyway e evesbly of e meo s no a majo ceon n coosng e meo. Hsoc noe: Te algom was fs use n 1791 by Delambe, an as been escovee many mes snce en, mos ecenly by Vele n 1960 s fo molecula ynamcs. I was also use by Cowell an Commeln n 1909 o compue e ob of Halley s come, an by Söme n 1907 o suy e moon of eleccal pacles n a magnec fel. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

14 Vele algom: A few small poblems + = F/m + O Te velocy oes no ene s algom explcly. We nee veloces n MD smulaons o compue knec enegy so a we can ceck consevaon of e oal enegy. Te nal conons ae ofen specfe as e nal coonaes an nal veloces, wle e Vele nvolves posons only. How o nouce nal empeaue? One can use v = [+ - -]/ o compue e veloces, bu e local eo assocae w s expesson s of oe ae an. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

15 Velocy Vele algom Anoe mplemenaon of e Vele algom a explcly nclues veloces s socalle Velocy Vele Algom. + = + v + F/ m + O v+ = v + F/m + F+/m/+ O Self-sang fom e posons an veloces a e nal me. Maemacally encal o e ognal Vele algom. Do no nee o soe e values of an v a wo ffeen mes. Noe a you o no nee sa-up calculaon of 0 - a s neee n e ognal Vele. Noe a you ave o calculae new foces F+ afe calculaon of new posons + bu befoe calculaon of new veloces. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

16 Peco-coeco algoms Anoe commonly use class of meos o negae euaons of moon ae peco-coeco algoms. Tese meos can be use n smulaons w velocy-epenen foces, suc as n consan-empeaue MD meos a wll be scusse lae n s couse. Peco. Fom e posons an e me evaves up o a cean oe all a me one pecs e same uanes a me + usng e Taylo expanson. Foce calculaon. Te foce acng on a gven pacle s compue fo e pece posons. Te acceleaon a = F/m wll be n geneal ffeen fom e pece acceleaon. Te ffeence beween e wo consues e eo sgnal. Coeco. Te eo sgnal s use o coec e posons an e evaves. All e coecons ae popoonal o e eo sgnal. Te coeffcens of popoonaly ae magc numbes cosen o maxmze e sably of e algom. Ts meo s use n many eal MD smulaons. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

17 Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle 1. Pece values Taylo expansons: mplemene n e MSE67-MD coe Noseck ff-oe peco-coeco algom Le us efne e followng vaables:

18 Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle mplemene n e MSE67-MD coe Noseck ff-oe peco-coeco algom. Eo n acceleaon: m F. Coecon smoong fle: C 0 c C 1 1 c 1 C c C c C c C 5 5 c 5 C 0 = /0 C 1 = 51/60 C = 1 C = 11/18 C = 1/6 C 5 = 1/60 Refeence: Allen & Tlesley, pp.7-75 an 0-1

19 Vele vs. Peco-Coeco fo MD Smulaons Velocy Vele an peco-coeco ae e mos effcen negaos fo MD. Peco-coeco meos: e man savanage s long-em enegy f e enegy eo s nceasng lnealy w me. ave vey goo local enegy consevaon small flucuaons. can be use w velocy-epenen foces e.g. w Nosé-Hoove meo fo consan empeaue smulaons. s easy o use w mulple mesep algoms. s no me-evesble. Vele meo ave a muc bee long-em enegy consevaon as compae o pecocoeco, ncease of e mesep ypcally esuls n lage local enegy flucuaons as compae o peco-coeco, bu no n enegy f. canno be use wen foces epen on e veloces of e aoms. s me evesble. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

20 Vele vs. Peco-Coeco fo MD Smulaons Toal enegy vaaon n a Lenna Jones LJ cubc sysem un cells, 08 aoms a T = 0.7ε ~T m, bu e sysem emans sol. Plos ae sown n LJ uns, e oal knec enegy s ~000ε. Wen even lage me sep s use, Gea5 pefoms muc wose an Vele. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

21 MSE 70/670 Refeences fo ODE 1. Wllam H. Pess, e al., Numecal Recpes n Foan 90 o C : Te a of scenfc an paallel compung, Cambge Unvesy Pess, s avalable fo fee n ps/pf fomas fom p:// M. P. Allen, D. J. Tlesley, Compue smulaon of lus Claenon Pess: Oxfo, 1990, pages 7-8 an 0- a goo escpon of meos fo soluon of onay ffeenal euaons use n molecula ynamcs Vele, peco-coeco.. D. Fenkel, B. Sm, Unesanng molecula smulaon fom algoms o applcaons Acaemc Pess: San Dego, 1996, pages moe on meos use n molecula ynamcs.. Dek Raabe, Compuaonal maeals scence: e smulaon of maeals, mcosucues an popees Wley-VCH: Wenem, 1998, pages 9-0 so noucon o ffeenal euaons. Unvesy of Vgna, MSE 70/670: Inoucon o Aomsc Smulaons, Leon Zgle

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma