###################################################################### # # A 'most robust' 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 robust" in the sense that # all weights b_i are non-negative, 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 norm as # # T72 ~ .00010103 # # (Formulas with slightly different nodes c_i can # have a slightly smaller error norm perhaps # achieved by having a larger maximum coefficient.) # # In addition, b_5=0, and this may be a benefit # for related methods for other types of problems. # # 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 ~ .0001428 # # This 2-norm has a single maximum value 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 ~ .00010103, # # the endpoint error of the propagating method. This # 2-norm has three local maximum values on [0,1]. # Attempts to make this error norm monotone increasing # over the interval [0,1] were unsuccessful. # # 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] = 9/50 c[3] = 1/6 c[4] = 1/4 c[5] = 53/100 c[6] = 3/5 c[7] = 4/5 c[8] = 1 c[9] = 1 # # ******************************************************** # COUPLING COEFFICIENTS # --------------------- #for c[1] = 0 # # for c[2] = 9/50 a[2,1] = 9/50 # # for c[3] = 1/6 a[3,1] = 29/324 a[3,2] = 25/324 # # for c[4] = 1/4 a[4,1] = 1/16 a[4,2] = 0 a[4,3] = 3/16 # # for c[5] = 53/100 a[5,1] = 79129/250000 a[5,2] = 0 a[5,3] = -261237/250000 a[5,4] = 19663/15625 # # for c[6] = 3/5 a[6,1] = 1336883/4909125 a[6,2] = 0 a[6,3] = -25476/30875 a[6,4] = 194159/185250 a[6,5] = 8225/78546 # # for c[7] = 4/5 a[7,1] = -2459386/14727375 a[7,2] = 0 a[7,3] = 19504/30875 a[7,4] = 2377474/13615875 a[7,5] = -6157250/5773131 a[7,6] = 902/735 # # for c[8] = 1 a[8,1] = 2699/7410 a[8,2] = 0 a[8,3] = -252/1235 a[8,4] = -1393253/3993990 a[8,5] = 236875/72618 a[8,6] = -135/49 a[8,7] = 15/22 # # for c[9] = 1 a[9,1] = 11/144 a[9,2] = 0 a[9,3] = 0 a[9,4] = 256/693 a[9,5] = 0 a[9,6] = 125/504 a[9,7] = 125/528 a[9,8] = 5/72 # # ******************************************************** # High order weights c[9] = 1 # (This is also the propagating stage 9, with a[9,i] = b[i].) # ----------------------------------------------------------- # b[1] = 11/144 b[2] = 0 b[3] = 0 b[4] = 256/693 b[5] = 0 b[6] = 125/504 b[7] = 125/528 b[8] = 5/72 b[9] = 0 # # ******************************************************** # Low order weights with c[extra] = 1 # -------------------------------------------------- # bh[1] = 28/477 bh[2] = 0 bh[3] = 0 bh[4] = 212/441 bh[5] = -312500/366177 bh[6] = 2125/1764 bh[7] = 0 bh[8] = -2105/35532 bh[9] = 2995/17766 # #*********************************************************** # # Largest coefficient in b or A ONLY has magnitude 3.261932 # #*********************************************************** # SUMMARY OF NORMS OF ERRORS: A71, A72, A7inf #---------------------------------------------------- # A_[7, 1] = .5274200310e-3 # A_[7, 2] = .1010283654e-3 # A_[7,oo] = .3869047619e-4 #**************************************************** # # 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 about [-4.324025812 , 0] # # Stability Boundaries of Low Order Method #---------------------------------------- # Real Stability Interval is about [-4.052296973 , 0] # ############################################################# # # ONE ADDITIONAL STAGE FOR INTERPOLANT OF ORDER 5 # # Coupling coefficients for c[10] = 1/2 # ---------------------------------------------------- a[10,1] = 1122414499/13386894336 a[10,2] = 0 a[10,3] = 0 a[10,4] = 1281083/3946608 a[10,5] = 41484375/162687952 a[10,6] = -184172125/884040192 a[10,7] = 4709875/84194304 a[10,8] = 1019015/126291456 a[10,9] = -67077/3508096 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi5 WITH 10 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi5[1] bi5[1,1] = 1 u bi5[1,2] = -1230040003/278893632 u^2 bi5[1,3] = 3949011217/418340448 u^3 bi5[1,4] = -2904068935/278893632 u^4 bi5[1,5] = 24431774/4357713 u^5 bi5[1,6] = -3331125/2905142 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] = 20224415/2110339 u^2 bi5[4,3] = -641415310/18993051 u^3 bi5[4,4] = 100989375/2110339 u^4 bi5[4,5] = -193181536/6331017 u^5 bi5[4,6] = 15228000/2110339 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[5] bi5[5,1] = 0 u bi5[5,2] = -99140625/10167997 u^2 bi5[5,3] = 760781250/10167997 u^3 bi5[5,4] = -255234375/1452571 u^4 bi5[5,5] = 1687500000/10167997 u^5 bi5[5,6] = -562500000/10167997 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[6] bi5[6,1] = 0 u bi5[6,2] = 256109125/18417504 u^2 bi5[6,3] = -2618512375/27626256 u^3 bi5[6,4] = 3887961625/18417504 u^4 bi5[6,5] = -110497750/575547 u^5 bi5[6,6] = 11896875/191849 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[7] bi5[7,1] = 0 u bi5[7,2] = -24780625/19294528 u^2 bi5[7,3] = 381456875/28941792 u^3 bi5[7,4] = -707528125/19294528 u^4 bi5[7,5] = 12166750/301477 u^5 bi5[7,6] = -9253125/602954u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[8] bi5[8,1] = 0 u bi5[8,2] = -1289445/877024 u^2 bi5[8,3] = 39045605/3946608 u^3 bi5[8,4] = -20496745/877024 u^4 bi5[8,5] = 1920590/82221 u^5 bi5[8,6] = -228625/27407u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi5[9] bi5[9,1] = 0 u bi5[9,2] = 312667/219256 u^2 bi5[9,3] = -1181783/109628 u^3 bi5[9,4] = 5965875/219256 u^4 bi5[9,5] = -788872/27407 u^5 bi5[9,6] = 299500/27407 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] = 19/20 # ---------------------------------------------------- a[11,1] = 51553925666573/669344716800000 a[11,2] = 0 a[11,3] = 0 a[11,4] = 5562317766977/15194440800000 a[11,5] = -41830875/2603007232 a[11,6] = 90705839393/353616076800 a[11,7] = 81497079563/370454937600 a[11,8] = 78900944453/1262914560000 a[11,9] = -5656570731/175404800000 a[11,10] = 3249/200000 # # ******************************************************** # # Coupling coefficients for c[12] = 9/50 # ---------------------------------------------------- a[12,1] = 8772752114029937691/112461115520000000000 a[12,2] = 0 a[12,3] = 0 a[12,4] = 935987914749411/5167855000000000 a[12,5] = 1871260623/33146434048 a[12,6] = 29592576132291/625404416000000 a[12,7] = 1105989441789861/9827783680000000 a[12,8] = 383940581029119/11167936000000000 a[12,9] = 992433017115603/13959920000000000 a[12,10] = -759789782621397/3489980000000000 a[12,11] = -6073879101561/33154810000000 # # -------------------------------------------------------- # COEFFICIENTS FOR INTERPOLANT bi6 WITH 12 STAGES # -------------------------------------------------------- # # COEFFICIENTS OF bi6[1] bi6[1,1] = 1 u bi6[1,2] = -160783/35568 u^2 bi6[1,3] = 395191/53352 u^3 bi6[1,4] = -1717/456 u^4 bi6[1,5] = -5117/4446 u^5 bi6[1,6] = 7375/6669 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] = 14592/1001 u^2 bi6[4,3] = -841216/9009 u^3 bi6[4,4] = 6656/33 u^4 bi6[4,5] = -538624/3003 u^5 bi6[4,6] = 512000/9009 u^6 + # -------------------------------------------------------- # # COEFFICIENTS OF bi6[5] bi6[5,1] = 0 u bi6[5,2] = 0 u^2 bi6[5,3] = 0 u^3 bi6[5,4] = 0 u^4 bi6[5,5] = 0 u^5 bi6[5,6] = 0 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[6] bi6[6,1] = 0 u bi6[6,2] = 7125/728 u^2 bi6[6,3] = -205375/3276 u^3 bi6[6,4] = 1625/12 u^4 bi6[6,5] = -32875/273 u^5 bi6[6,6] = 31250/819 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[7] bi6[7,1] = 0 u bi6[7,2] = 21375/2288 u^2 bi6[7,3] = -205375/3432 u^3 bi6[7,4] = 11375/88 u^4 bi6[7,5] = -32875/286 u^5 bi6[7,6] = 15625/429 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[8] bi6[8,1] = 0 u bi6[8,2] = 285/104 u^2 bi6[8,3] = -8215/468 u^3 bi6[8,4] = 455/12 u^4 bi6[8,5] = -1315/39 u^5 bi6[8,6] = 1250/117 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[9] bi6[9,1] = 0 u bi6[9,2] = 3933/533 u^2 bi6[9,3] = -77801/1599 u^3 bi6[9,4] = 4554/41 u^4 bi6[9,5] = -56868/533 u^5 bi6[9,6] = 59000/1599 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[10] bi6[10,1] = 0 u bi6[10,2] = -2033/104 u^2 bi6[10,3] = 171355/1404 u^3 bi6[10,4] = -18173/72 u^4 bi6[10,5] = 25372/117 u^5 bi6[10,6] = -23500/351 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[11] bi6[11,1] = 0 u bi6[11,2] = -328000/19019 u^2 bi6[11,3] = 57712000/513513 u^3 bi6[11,4] = -472000/1881 u^4 bi6[11,5] = 40000000/171171 u^5 bi6[11,6] = -40000000/513513 u^6 # -------------------------------------------------------- # # COEFFICIENTS OF bi6[12] bi6[12,1] = 0 u bi6[12,2] = -7421875/2954952 u^2 bi6[12,3] = 178515625/4432428 u^3 bi6[12,4] = -390625/3608 u^4 bi6[12,5] = 39062500/369369 u^5 bi6[12,6] = -39062500/1108107 ^6 # #******************************************************** # Norms of low order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- 0.1000000000 .2514988881e-4 .5334895445e-4 0.2000000000 .5487452962e-4 .1087206961e-3 0.3000000000 .6869260090e-4 .1260663317e-3 0.4000000000 .6947290200e-4 .1356772529e-3 0.5000000000 .6805941112e-4 .1428384993e-3 0.6000000000 .6947290200e-4 .1356772529e-3 0.7000000000 .6869260090e-4 .1260663317e-3 0.8000000000 .5487452962e-4 .1087206961e-3 0.9000000000 .2514988881e-4 .5334895445e-4 1.0000000000 -.2777777778e-40 .3044835995e-40 1.1000000000 .9826564848e-4 .1667556807e-3 1.2000000000 .6555836277e-3 .1068455184e-2 1.3000000000 .2303373923e-2 .3661871781e-2 1.4000000000 .6086218872e-2 .9518473596e-2 1.5000000000 .1359730066e-1 .2102425454e-1 1.6000000000 .2713189713e-1 .4160639285e-1 1.7000000000 .4985893597e-1 .7598664787e-1 1.8000000000 .8601060718e-1 .1304614610 1.9000000000 .1410900339 .2132087471 2.0000000000 .2220970015 .3346213710 # #******************************************************** # Norms of high order INTERPOLANT coefficients on [0,2] # u Max norm 2-norm #------------------------------------------------- 0.1000000000 -.1748614393e-4 .3871244616e-4 0.2000000000 -.2540078144e-4 .5590264830e-4 0.3000000000 -.1058820742e-4 .2332485071e-4 0.4000000000 -.2131363334e-4 .3268258460e-4 0.5000000000 -.3428310604e-4 .5783268405e-4 0.6000000000 -.3119605679e-4 .5357322383e-4 0.7000000000 .2557288565e-4 .5242810116e-4 0.8000000000 .3293130242e-4 .7854022080e-4 0.9000000000 .3771221719e-4 .9853190954e-4 1.0000000000 .3869047619e-4 .1010283654e-3 1.1000000000 .4190182635e-4 .1180807274e-3 1.2000000000 .1403526331e-3 .3002636971e-3 1.3000000000 .4916601709e-3 .1084305465e-2 1.4000000000 .1379503521e-2 .3334253336e-2 1.5000000000 .3261475687e-2 .8671876200e-2 1.6000000000 .7028522575e-2 .1988446987e-1 1.7000000000 .1468930269e-1 .4144286253e-1 1.8000000000 .2839868211e-1 .8015256605e-1 1.9000000000 .5159999217e-1 .1459511745 2.0000000000 .8911898503e-1 .2528661715 # ********************************************************