###################################################################### # # A 'most efficient' Runge--Kutta (9,6(5)) pair # # These are exact RATIONAL coefficients for # # a NINE-stage FSAL pair of methods of # orders p=6 and p=5, with # dominant stage order = 3, # # together with exact RATIONAL coefficients # # for two interpolants of orders 5 and 6 # # which require 1 and 2 extra stages respectively. # (Companion files list only the RATIONAL and 40-digit # floating point approximations respectively.) # # This procedure is "most efficient" in the sense that # the maximum coefficient within b and A is not # large, and it has a propagating formula which almost # minimizes the 2-norm of the local truncation error as # # T72 ~ .00000144 # # (Formulas with slightly different nodes c_i can # have a slightly smaller error norm perhaps # achieved by having a larger maximum coefficient.) # # Additional stages and interpolating weights allow # for the computation of an approximation at any # point of the domain of solution of order up to p. # These interpolants have continuous derivatives. # # The additional node c[10] = 1/2 is chosen to achieve # order 5 interpolation in ten stages. In the interval # [0,1], the 2-norm of the local truncation error of # this interpolant has a maximum 2-norm of # # T_62 ~ .000318 # # This 2-norm has three local maximum values on [0,1]. # # The remaining two nodes are selected in an attempt to # minimize the maximum of the 2-norm of the local # truncation error on the interval [0,1] for the # interpolant of order 6, and this value is # # T_72 ~ .0000444 # # This 2-norm has two local maximum values on [0,1]. # Attempts to make this error norm monotone over the # interval [0,1] were unsuccessful. (See end of file.) # # The formulas scanned for this optimal formula include # those developed in J.H. Verner, SIAM NA 1978, 772-790, # "Explicit Runge--Kutta methods with estimates of the # Local Truncation Error". The pair selected is one from # the pairs in J.H. Verner, Annals of Num. Math 1 1994, # 225-244, "Strategies for deriving new explicit Runge-- # Kutta pairs", which require only ten stages, but solve # the order conditions in a different way. It is possible, # that one of the 11-stage contemporary or 12-stage FSAL # methods derived by Sharp and Verner, SIAM NA 31, 1994, # 1169--1190, "Completely imbedded Runge--Kutta pairs" # may yield particular pairs of equivalant or more # efficiency. # ###################################################################### # # NODES # ----- c[1] = 0 c[2] = 3/50 c[3] = 1439/15000 c[4] = 1439/10000 c[5] = 4973/10000 c[6] = 389/400 c[7] = 1999/2000 c[8] = 1 c[9] = 1 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- #for c[1] = 0 # # for c[2] = 3/50 a[2,1] = 3/50 # # for c[3] = 1439/15000 a[3,1] = 519479/27000000 a[3,2] = 2070721/27000000 # # for c[4] = 1439/10000 a[4,1] = 1439/40000 a[4,2] = 0 a[4,3] = 4317/40000 # # for c[5] = 4973/10000 a[5,1] = 109225017611/82828840000 a[5,2] = 0 a[5,3] = -417627820623/82828840000 a[5,4] = 43699198143/10353605000 # # for c[6] = 389/400 a[6,1] = -8036815292643907349452552172369/ 191934985946683241245914401600 a[6,2] = 0 a[6,3] = 246134619571490020064824665/1543816496655405117602368 a[6,4] = -13880495956885686234074067279/ 113663489566254201783474344 a[6,5] = 755005057777788994734129/136485922925633667082436 # # for c[7] = 1999/2000 a[7,1] = -1663299841566102097180506666498880934230261/ 30558424506156170307020957791311384232000 a[7,2] = 0 a[7,3] = 130838124195285491799043628811093033/ 631862949514135618861563657970240 a[7,4] = -3287100453856023634160618787153901962873/ 20724314915376755629135711026851409200 a[7,5] = 2771826790140332140865242520369241/ 396438716042723436917079980147600 a[7,6] = -1799166916139193/96743806114007800 # # for c[8] = 1 a[8,1] = -832144750039369683895428386437986853923637763/ 15222974550069600748763651844667619945204887 a[8,2] = 0 a[8,3] = 818622075710363565982285196611368750/ 3936576237903728151856072395343129 a[8,4] = -9818985165491658464841194581385463434793741875/ 61642597962658994069869370923196463581866011 a[8,5] = 31796692141848558720425711042548134769375/ 4530254033500045975557858016006308628092 a[8,6] = -14064542118843830075/766928748264306853644 a[8,7] = -1424670304836288125/2782839104764768088217 # # for c[9] = 1 a[9,1] = 382735282417/11129397249634 a[9,2] = 0 a[9,3] = 0 a[9,4] = 5535620703125000/21434089949505429 a[9,5] = 13867056347656250/32943296570459319 a[9,6] = 626271188750/142160006043 a[9,7] = -51160788125000/289890548217 a[9,8] = 163193540017/946795234 # # ******************************************************** # High order weights c[9] = 1 # (This is also the propagating stage 9, with a[9,i] = b[i].) # -------------------------------------------------------- # b[1] = 382735282417/11129397249634 b[2] = 0 b[3] = 0 b[4] = 5535620703125000/21434089949505429 b[5] = 13867056347656250/32943296570459319 b[6] = 626271188750/142160006043 b[7] = -51160788125000/289890548217 b[8] = 163193540017/946795234 b[9] = 0 # # ******************************************************** # Low order weights with c[extra] = 1 # -------------------------------------------------- # bh[1] = 124310637869885675646798613/2890072468789466426596827670 bh[2] = 0 bh[3] = 0 bh[4] = 265863151737164990361330921875/ 1113197271463372303940319369579 bh[5] = 3075493557174030806536302953125/ 6843749922042323876546949699876 bh[6] = 67798000008733879813263055/29532792147666737550036372 bh[7] = -1099436585155390846238326375/15055706496446408859196167 bh[8] = 26171252653086373181571802/368794478890732346033505 bh[9] = 1/30 # #************************************************************ # # Largest coefficient in b or A ONLY has magnitude 207.9528 # #************************************************************ # SUMMARY OF NORMS OF ERRORS: A71, A72, A7inf #---------------------------------------------------- # A_[7,1] = .5384213684e-5 # A_[7,2] = .1446174055e-5 # A_[7,oo] = .8692258727e-6 #**************************************************** # # END OF GENERATION OF THE PAIR OF RK METHODS # ############################################################# # # START OF GENERATION OF STABILITY INTERVALS # ############################################################# # # Stability Boundaries of High Order Method # ----------------------------------------- # Real Stability Interval is nearly [ -4.855274314, 0] # # Stability Boundaries of Low Order Method # ---------------------------------------- # Real Stability Interval is nearly [ -4.386141682, 0] # ############################################################# # # ONE ADDITIONAL STAGE FOR INTERPOLANT OF ORDER 5 # # Coupling coefficients for c[10] = 1/2 # ---------------------------------------------------- a[10,1] = 35289331988986254405692535758830683/ 2135620454874580332949729350544993288 a[10,2] = 0 a[10,3] = 0 a[10,4] = 313937014583068512255490687992212890625/ 1028247080705354654473994781524199691557 a[10,5] = 1309307687253621245836726130885318359375/ 6321490412177191231557635904400612215708 a[10,6] = -35295844079877524186147726060781875/ 27279088881521314684841470427640876 a[10,7] = 794353492803973228770716697389421875/ 13906777037439977359946774228636361 a[10,8] = -15228408956329265381787438679500067/ 272520859345009876882656783678732 a[10,9] = 28587810357600962662801/1151340224617184234295192 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi5 WITH 10 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi5[1] bi5[1,1] = 1 u bi5[1,2] = -2834058897718490495086218793721472473/ 533905113718645083237432337636248322 u^2 bi5[1,3] = 2718025628974094767211106485595747024/ 266952556859322541618716168818124161 u^3 bi5[1,4] = -2007493102587435133656511668645819580/ 266952556859322541618716168818124161 u^4 bi5[1,5] = 249346645146318025711596899739877112/ 266952556859322541618716168818124161 u^5 bi5[1,6] = 199378106425839009650374224000000000/ 266952556859322541618716168818124161 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[2] bi5[2,1] = 0 u bi5[2,2] = 0 u^2 bi5[2,3] = 0 u^3 bi5[2,4] = 0 u^4 bi5[2,5] = 0 u^5 bi5[2,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[3] bi5[3,1] = 0 u bi5[3,2] = 0 u^2 bi5[3,3] = 0 u^3 bi5[3,4] = 0 u^4 bi5[3,5] = 0 u^5 bi5[3,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[4] bi5[4,1] = 0 u bi5[4,2] = 2149739120967678287896284375471359375000/ 342749026901784884824664927174733230519 u^2 bi5[4,3] = -5492958105397111152037592078122406250000/ 342749026901784884824664927174733230519 u^3 bi5[4,4] = 4402390631408885178088267799125656250000/ 342749026901784884824664927174733230519 u^4 bi5[4,5] = -56249742645646759461666034659375000000/ 48964146700254983546380703882104747217 u^5 bi5[4,6] = -1730712729390963625437276250000000000000/ 1028247080705354654473994781524199691557 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[5] bi5[5,1] = 0 u bi5[5,2] = 3622473030746576800982284292813464843750/ 526790867681432602629802992033384351309 u^2 bi5[5,3] = -38933691634017210049674664360163992187500/ 1580372603044297807889408976100153053927 u^3 bi5[5,4] = 17495139028182305773126471135900867187500/ 526790867681432602629802992033384351309 u^4 bi5[5,5] = -9216003564492900591852706378813281250000/ 526790867681432602629802992033384351309 u^5 bi5[5,6] = 432678182347740906359319062500000000000/ 175596955893810867543267664011128117103 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[6] bi5[6,1] = 0 u bi5[6,2] = -80801688121532406876813280779008750/ 2273257406793442890403455868970073 u^2 bi5[6,3] = 1130047284618441598167544907799477500/ 6819772220380328671210367606910219 u^3 bi5[6,4] = -876257846328841227135521923123077500/ 2273257406793442890403455868970073 u^4 bi5[6,5] = 1005762761452595148951569429250170000/ 2273257406793442890403455868970073 u^5 bi5[6,6] = -138457018351277090034982100000000000/ 757752468931147630134485289656691 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[7] bi5[7,1] = 0 u bi5[7,2] = 8894101767966865321149886325974625000/ 4635592345813325786648924742878787 u^2 bi5[7,3] = -128889699381092513087660977440685250000/ 13906777037439977359946774228636361 u^3 bi5[7,4] = 96690747476972701449592103439602750000/ 4635592345813325786648924742878787 u^4 bi5[7,5] = -104976825419006157083194997793935000000/ 4635592345813325786648924742878787 u^5 bi5[7,6] = 13845701835127709003498210000000000000/ 1545197448604441928882974914292929 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[8] bi5[8,1] = 0 u bi5[8,2] = -85529300113974351208051144641213185/ 45420143224168312813776130613122 u^2 bi5[8,3] = 620054801234124026686518242620266725/ 68130214836252469220664195919683 u^3 bi5[8,4] = -464947578142702593618050980843076970/ 22710071612084156406888065306561 u^4 bi5[8,5] = 72055052308090805849478606896950124/ 3244295944583450915269723615223 u^5 bi5[8,6] = -598331009666258752869007208000000000/ 68130214836252469220664195919683 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[9] bi5[9,1] = 0 u bi5[9,2] = 5709918156918632012901/47972509359049343095633 u^2 bi5[9,3] = -17993572040875704216709/143917528077148029286899 u^3 bi5[9,4] = 85388999974381230343470/47972509359049343095633 u^4 bi5[9,5] = -223596609894610627617468/47972509359049343095633 u^5 bi5[9,6] = 415486647330808000000000/143917528077148029286899 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[10] bi5[10,1] = 0 u bi5[10,2] = -8 u^2 bi5[10,3] = 32 u^3 bi5[10,4] = -40 u^4 bi5[10,5] = 16 u^5 bi5[10,6] = 0 u^6 # # ******************************************************** # # TWO ADDITIONAL STAGES FOR INTERPOLANT OF ORDER 6 # # Coupling coefficients for c[11] = 207/250 # ---------------------------------------------------- a[11,1] = 2486392061981208591025761263164027224438868971/ 65173964076983042387381877152862343994140625000 a[11,2] = 0 a[11,3] = 0 a[11,4] = 2330654500023704838558579323179918419669/ 9313832252765893609365894760182968220625 a[11,5] = 5283259505481013273874688940942473187741/ 16258977397575080328080339260289640472500 a[11,6] = 9989685106081485386057729811605187743723/ 5481427003263510055949691042076757812500 a[11,7] = -65815640423883764662985178413751186161/ 971969007022721623945108012714453125 a[11,8] = 183066350554023250298437927498791289370414247/ 2772225538584491748887703284492309570312500 a[11,9] = -426178927623072052719640507155669/ 11712038417736656029207275390625000 a[11,10] = 3248339841/30517578125 # # ******************************************************** # # Coupling coefficients for c[12] = 7/25 # ---------------------------------------------------- a[12,1] = 4676747786898097735038451956075910033997933945857/ 41838231186922043164464169766109251031526972656250 a[12,2] = 0 a[12,3] = 0 a[12,4] = 1320032412954312695441306548681592444623240/ 51248457773784347881352490499724836575577977 a[12,5] = 2087002134582726310861746540254017903014374710/ 551367099344274428347227263044005314054687829 a[12,6] = 3432932836484348829479408524345545011748570706/ 37176735450871998946806722732624135633015625 a[12,7] = -2316434358511265475362584844804601519943610264/ 606481922490173339581866127622363581143375 a[12,8] = 82514605285282414051716141603447021470923168793/ 22107104196177512751528507591142367597656250 a[12,9] = -7560161019374651900153317984708038834/ 7028170531590816328729091157353515625 a[12,10] = -21655450552377696842870155771710589332/ 6701278878958685336695179940732421875 a[12,11] = -3194830887993202085244614477336220/ 678662636676110315314332975245759 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 12 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1 u bi6[1,2] = -940811006205413129/120948724610397495 u^2 bi6[1,3] = 88342864458754360181/3265615564480732365 u^3 bi6[1,4] = -99667000922033025307/2177077042987154910 u^4 bi6[1,5] = 7995049273203130972/217707704298715491 u^5 bi6[1,6] = -7303903485456272500/653123112896146473 u^6 + # -------------------------------------------------------- # # COEFFICIENTS OF bi6[2] bi6[2,1] = 0 u bi6[2,2] = 0 u^2 bi6[2,3] = 0 u^3 bi6[2,4] = 0 u^4 bi6[2,5] = 0 u^5 bi6[2,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[3] bi6[3,1] = 0 u bi6[3,2] = 0 u^2 bi6[3,3] = 0 u^3 bi6[3,4] = 0 u^4 bi6[3,5] = 0 u^5 bi6[3,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[4] bi6[4,1] = 0 u bi6[4,2] = 2214248281250000/133130993475189 u^2 bi6[4,3] = -49918013252500000000/578720428636646583 u^3 bi6[4,4] = 1440368506953125000/8387252588936907 u^4 bi6[4,5] = -28873797587500000000/192906809545548861 u^5 bi6[4,6] = 27678103515625000000/578720428636646583 u^6 + # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0 u bi6[5,2] = 893038428789062500/32943296570459319 u^2 bi6[5,3] = -125047567320625000000/889469007402401613 u^3 bi6[5,4] = 82988785418183593750/296489669134133871 u^4 bi6[5,5] = -72330565909375000000/296489669134133871 u^5 bi6[5,6] = 69335281738281250000/889469007402401613 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0 u bi6[6,2] = 40331864555500/142160006043 u^2 bi6[6,3] = -5647463071672000/3838320163161 u^3 bi6[6,4] = 3747982556193250/1279440054387 u^4 bi6[6,5] = -3266630520520000/1279440054387 u^5 bi6[6,6] = 3131355943750000/3838320163161 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0 u bi6[7,2] = -143250206750000/12603936879 u^2 bi6[7,3] = 461347522996000000/7827044801859 u^3 bi6[7,4] = -13312037070125000/113435431911 u^4 bi6[7,5] = 266854670860000000/2609014933953 u^5 bi6[7,6] = -255803940625000000/7827044801859 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0 u bi6[8,2] = 3753451420391/338141155 u^2 bi6[8,3] = -3679035166143248/63908678295 u^3 bi6[8,4] = 4883240297928691/42605785530 u^4 bi6[8,5] = -425608752364336/4260578553 u^5 bi6[8,6] = 407983850042500/12781735659 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0 u bi6[9,2] = -69713/23220 u^2 bi6[9,3] = 4685161/313470 u^3 bi6[9,4] = -135239/4860 u^4 bi6[9,5] = 228046/10449 u^5 bi6[9,6] = -186250/31347 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[10] bi6[10,1] = 0 u bi6[10,2] = -132664/6765 u^2 bi6[10,3] = 17011336/182655 u^3 bi6[10,4] = -10067296/60885 u^4 bi6[10,5] = 1579832/12177 u^5 bi6[10,6] = -1385000/36531 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[11] bi6[11,1] = 0 u bi6[11,2] = -2734375000/149990751 u^2 bi6[11,3] = 391796875000/4049750277 u^3 bi6[11,4] = -6250000000/31393413 u^4 bi6[11,5] = 244140625000/1349916759 u^5 bi6[11,6] = -244140625000/4049750277 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0 u bi6[12,2] = -15453125/1139292 u^2 bi6[12,3] = 1393796875/15380442 u^3 bi6[12,4] = -2092203125/10253628 u^4 bi6[12,5] = 488281250/2563407 u^5 bi6[12,6] = -488281250/7690221 u^6 # #******************************************************** # Norms of low order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- 0.1000000000 0.1167749412e-3 0.1440034813e-3 0.2000000000 0.1614240810e-3 0.1990071227e-3 0.3000000000 0.2258071794e-4 0.2771805088e-4 0.4000000000 -0.1707334172e-3 0.2108327188e-3 0.5000000000 -0.2576775359e-3 0.3180913837e-3 0.6000000000 -0.1707334172e-3 0.2108327188e-3 0.7000000000 0.2258071794e-4 0.2771805088e-4 0.8000000000 0.1614240810e-3 0.1990071227e-3 0.9000000000 0.1167749412e-3 0.1440034813e-3 1.0000000000 0.8333333333e-40 0.1085608366e-39 1.1000000000 0.4548515160e-3 0.5609871807e-3 1.2000000000 0.3032892179e-2 0.3740683804e-2 1.3000000000 0.1065234774e-1 0.1313850720e-1 1.4000000000 0.2814038736e-1 0.3470844789e-1 1.5000000000 0.6285883179e-1 0.7753084811e-1 1.6000000000 0.1254132892 0.1546870842 1.7000000000 0.2304457188 0.2842371526 1.8000000000 0.3975104221 0.4903001602 1.9000000000 0.6520334625 0.8042377177 2.0000000000 1.026355512 1.265940238 # #******************************************************** # Norms of high order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- .1000000000 0.1346683887e-4 0.1793720883e-4 .2000000000 0.3187283885e-4 0.4000741709e-4 .3000000000 0.3558253200e-4 0.4443863622e-4 .4000000000 0.2107729733e-4 0.2881895713e-4 .5000000000 -0.6959426407e-5 0.1733198141e-4 .6000000000 -0.2607320712e-4 0.3363473290e-4 .7000000000 -0.3503132089e-4 0.4430137891e-4 .8000000000 -0.2730149523e-4 0.4412093466e-4 .9000000000 -0.1009228752e-4 0.2499485587e-4 1.000000000 -0.8692258727e-6 0.1446174055e-5 1.100000000 0.4293123039e-4 0.1029659823e-3 1.200000000 0.3137394462e-3 0.7446300819e-3 1.300000000 0.1203049165e-2 0.2847492510e-2 1.400000000 0.3451002685e-2 0.8168285272e-2 1.500000000 0.8326443824e-2 0.1972815141e-1 1.600000000 0.1785620237e-1 0.4236329742e-1 1.700000000 0.3511054664e-1 0.8341079832e-1 1.800000000 0.6455080426e-1 0.1535439721 1.900000000 0.1124451510 0.2677717506 2.000000000 0.1873585679 0.4466164763 #********************************************************