FLINT 0.9.9
Fortran Library for numerical INTegration of differential equations
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Variables
butchertableaus Module Reference

Butcher Tableau Module. More...

Variables

integer, parameter erk_maxstages = 21
 Maximum number of stages for storage allocation.
 
integer, parameter erk_maxipdegree = 8
 
integer, parameter erk_max_a = int((ERK_MAXSTAGES-1)/2.0*ERK_MAXSTAGES)
 
integer doi
 
integer, parameter dop54_q = 4
 min(p, phat) used in step-size computation
 
integer, parameter dop54_p = 5
 Order of the propagating stage.
 
integer, parameter dop54_phat = 4
 Order of the error stage.
 
integer, parameter dop54_pstar = 4
 Order of interpolants.
 
integer, parameter dop54_s = 7
 Total Number of stages.
 
logical, parameter dop54_fsal = .TRUE.
 Is this a FSAL method?
 
integer, parameter dop54_sint = 7
 Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.
 
real(wp), parameter dop54_beta = 0.04_WP
 Lund stabilization parameters: beta and beta_multiplier. Positive values (< 0.1) make the step size control more stable. See Hairer's original DOPRI5 codes.
 
real(wp), parameter dop54_beta_mult = 0.75_WP
 
real(wp), dimension(2:*), parameter dop54_c = [ 1.0_WP/5.0_WP, 3.0_WP/10.0_WP, 4.0_WP/5.0_WP, 8.0_WP/9.0_WP, 1.0_WP, 1.0_WP ]
 c_i where i = 2 to s, s=number of stages
 
real(wp), dimension(*), parameter dop54_b = [ 35.0_WP/384.0_WP, 0.0_WP, 500.0_WP/1113.0_WP, 125.0_WP/192.0_WP, -2187.0_WP/6784.0_WP, 11.0_WP/84.0_WP, 0.0_WP ]
 b_i where i = 1 to sint, sint=number of main stages These are higher order weights
 
real(wp), dimension(*), parameter dop54_bh = [ 5179.0_WP/57600.0_WP, 0.0_WP, 7571.0_WP/16695.0_WP, 393.0_WP/640.0_WP, -92097.0_WP/339200.0_WP, 187.0_WP/2100.0_WP, 1.0_WP/40.0_WP ]
 bh_i where i = 1 to sint These are lower order weights for error estimation in local extrapolation case
 
real(wp), dimension(*), parameter dop54_e = DOP54_b - DOP54_bh
 e_i where i = 1 to sint
 
real(wp), dimension(1:*), parameter dop54_a = [ 1.0_WP/5.0_WP, 3.0_WP/40.0_WP, 9.0_WP/40.0_WP, 44.0_WP/45.0_WP, -56.0_WP/15.0_WP, 32.0_WP/9.0_WP, 19372.0_WP/6561.0_WP, -25360.0_WP/2187.0_WP, 64448.0_WP/6561.0_WP, -212.0_WP/729.0_WP, 9017.0_WP/3168.0_WP, -355.0_WP/33.0_WP, 46732.0_WP/5247.0_WP, 49.0_WP/176.0_WP, -5103.0_WP/18656.0_WP, DOP54_b(1:6) ]
 a_ij, where i = 2 to s, j = 1 to i-1.
 
integer, dimension(*), parameter dop54_di_nz = [1,3,4,5,6,7]
 Coefficients for 4th-order interpolation.
 
real(wp), dimension(*, 1:*), parameter dop54_d = reshape([ -12715105075.0_WP/11282082432.0_WP, 87487479700.0_WP/32700410799.0_WP, -10690763975.0_WP/1880347072.0_WP, 701980252875.0_WP/199316789632.0_WP, -1453857185.0_WP/822651844.0_WP, 69997945.0_WP/29380423.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP ], [size(DOP54_di_NZ),2])
 dij, where i = 1 to s, j = 1 to pstar for a generic method
 
integer, parameter dop853_q = 7
 min(p, phat) used in step-size computation
 
integer, parameter dop853_p = 8
 Order of the propagating stage.
 
integer, parameter dop853_phat = 6
 Order of the error stage.
 
integer, parameter dop853_pstar = 7
 number of non-zero interpolation coefficients
 
integer, parameter dop853_s = 16
 Total Number of stages.
 
logical, parameter dop853_fsal = .TRUE.
 Is this a FSAL method?
 
integer, parameter dop853_sint = 13
 Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.
 
real(wp), parameter dop853_beta = 0.0_WP
 Lund stabilization parameters: beta and beta_multiplier. Positive values (< 0.1) make the step size control more stable. See Hairer's original DOPRI5 codes.
 
real(wp), parameter dop853_beta_mult = 0.2_WP
 
real(wp), dimension(2:*), parameter dop853_c = [ 0.526001519587677318785587544488e-01_WP, 0.789002279381515978178381316732e-01_WP, 0.118350341907227396726757197510e+00_WP, 0.281649658092772603273242802490e+00_WP, 0.333333333333333333333333333333e+00_WP, 0.25e+00_WP, 0.307692307692307692307692307692e+00_WP, 0.651282051282051282051282051282e+00_WP, 0.6e+00_WP, 0.857142857142857142857142857142e+00_WP, 1.0_WP, 1.0_WP, 0.1e+00_WP, 0.2e+00_WP, 0.777777777777777777777777777778e+00_WP ]
 c_i where i = 2 to s, s=number of stages
 
real(wp), dimension(*), parameter dop853_b = [ 5.42937341165687622380535766363e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 4.45031289275240888144113950566e0_WP, 1.89151789931450038304281599044e0_WP, -5.8012039600105847814672114227e0_WP, 3.1116436695781989440891606237e-1_WP, -1.52160949662516078556178806805e-1_WP, 2.01365400804030348374776537501e-1_WP, 4.47106157277725905176885569043e-2_WP, 0.0_WP ]
 b_i where i = 1 to sint, sint=number of main stages
 
real(wp), dimension(*), parameter dop853_bhh = [ 0.244094488188976377952755905512e+00_WP, 0.733846688281611857341361741547e+00_WP, 0.220588235294117647058823529412e-01_WP ]
 bhh_i where i = 1 to 3 (DOP853 3rd order corrector for 5th-order error estimator)
 
real(wp), dimension(*), parameter dop853_e = [ 0.1312004499419488073250102996e-01_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -0.1225156446376204440720569753e+01_WP, -0.4957589496572501915214079952e+00_WP, 0.1664377182454986536961530415e+01_WP, -0.3503288487499736816886487290e+00_WP, 0.3341791187130174790297318841e+00_WP, 0.8192320648511571246570742613e-01_WP, -0.2235530786388629525884427845e-01_WP, 0.0_WP ]
 e_i where i = 1 to sint
 
real(wp), dimension(1:*), parameter dop853_a = [ 5.26001519587677318785587544488e-2_WP, 1.97250569845378994544595329183e-2_WP, 5.91751709536136983633785987549e-2_WP, 2.95875854768068491816892993775e-2_WP, 0.0_WP, 8.87627564304205475450678981324e-2_WP, 2.41365134159266685502369798665e-1_WP, 0.0_WP, -8.84549479328286085344864962717e-1_WP, 9.24834003261792003115737966543e-1_WP, 3.7037037037037037037037037037e-2_WP, 0.0_WP, 0.0_WP, 1.70828608729473871279604482173e-1_WP, 1.25467687566822425016691814123e-1_WP, 3.7109375e-2_WP, 0.0_WP, 0.0_WP, 1.70252211019544039314978060272e-1_WP, 6.02165389804559606850219397283e-2_WP, -1.7578125e-2_WP, 3.70920001185047927108779319836e-2_WP, 0.0_WP, 0.0_WP, 1.70383925712239993810214054705e-1_WP, 1.07262030446373284651809199168e-1_WP, -1.53194377486244017527936158236e-2_WP, 8.27378916381402288758473766002e-3_WP, 6.24110958716075717114429577812e-1_WP, 0.0_WP, 0.0_WP, -3.36089262944694129406857109825e0_WP, -8.68219346841726006818189891453e-1_WP, 2.75920996994467083049415600797e1_WP, 2.01540675504778934086186788979e1_WP, -4.34898841810699588477366255144e1_WP, 4.77662536438264365890433908527e-1_WP, 0.0_WP, 0.0_WP, -2.48811461997166764192642586468e0_WP, -5.90290826836842996371446475743e-1_WP, 2.12300514481811942347288949897e1_WP, 1.52792336328824235832596922938e1_WP, -3.32882109689848629194453265587e1_WP, -2.03312017085086261358222928593e-2_WP, -9.3714243008598732571704021658e-1_WP, 0.0_Wp, 0.0_WP, 5.18637242884406370830023853209e0_WP, 1.09143734899672957818500254654e0_WP, -8.14978701074692612513997267357e0_WP, -1.85200656599969598641566180701e1_WP, 2.27394870993505042818970056734e1_WP, 2.49360555267965238987089396762e0_WP, -3.0467644718982195003823669022e0_WP, 2.27331014751653820792359768449e0_WP, 0.0_WP, 0.0_WP, -1.05344954667372501984066689879e1_WP, -2.00087205822486249909675718444e0_WP, -1.79589318631187989172765950534e1_WP, 2.79488845294199600508499808837e1_WP, -2.85899827713502369474065508674e0_WP, -8.87285693353062954433549289258e0_WP, 1.23605671757943030647266201528e1_WP, 6.43392746015763530355970484046e-1_WP, DOP853_b(1:12), 5.61675022830479523392909219681e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 2.53500210216624811088794765333e-1_WP, -2.46239037470802489917441475441e-1_WP, -1.24191423263816360469010140626e-1_WP, 1.5329179827876569731206322685e-1_WP, 8.20105229563468988491666602057e-3_WP, 7.56789766054569976138603589584e-3_WP, -8.298e-3_WP, 3.18346481635021405060768473261e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 2.83009096723667755288322961402e-2_WP, 5.35419883074385676223797384372e-2_WP, -5.49237485713909884646569340306e-2_WP, 0.0_WP, 0.0_WP, -1.08347328697249322858509316994e-4_WP, 3.82571090835658412954920192323e-4_WP, -3.40465008687404560802977114492e-4_WP, 1.41312443674632500278074618366e-1_WP, -4.28896301583791923408573538692e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -4.69762141536116384314449447206e0_WP, 7.68342119606259904184240953878e0_WP, 4.06898981839711007970213554331e0_WP, 3.56727187455281109270669543021e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, -1.39902416515901462129418009734e-3_WP, 2.9475147891527723389556272149e0_WP, -9.15095847217987001081870187138e0_WP ]
 a_ij, where i = 2 to s, j = 1 to i-1.
 
integer i
 Coefficients for 7th-order interpolation.
 
integer, dimension(*), parameter dop853_di_nz = [ 1, (i, i=6,16)]
 Non-zero coefficients for non-zero stages.
 
integer, dimension(*), parameter dop853_dj_nz = [4,5,6,7]
 
real(wp), dimension(*, 4:*), parameter dop853_d = reshape([ -0.84289382761090128651353491142e+01_WP, 0.56671495351937776962531783590e+00_WP, -0.30689499459498916912797304727e+01_WP, 0.23846676565120698287728149680e+01_WP, 0.21170345824450282767155149946e+01_WP, -0.87139158377797299206789907490e+00_WP, 0.22404374302607882758541771650e+01_WP, 0.63157877876946881815570249290e+00_WP, -0.88990336451333310820698117400e-01_WP, 0.18148505520854727256656404962e+02_WP, -0.91946323924783554000451984436e+01_WP, -0.44360363875948939664310572000e+01_WP, 0.10427508642579134603413151009e+02_WP, 0.24228349177525818288430175319e+03_WP, 0.16520045171727028198505394887e+03_WP, -0.37454675472269020279518312152e+03_WP, -0.22113666853125306036270938578e+02_WP, 0.77334326684722638389603898808e+01_WP, -0.30674084731089398182061213626e+02_WP, -0.93321305264302278729567221706e+01_WP, 0.15697238121770843886131091075e+02_WP, -0.31139403219565177677282850411e+02_WP, -0.93529243588444783865713862664e+01_WP, 0.35816841486394083752465898540e+02_WP, 0.19985053242002433820987653617e+02_WP, -0.38703730874935176555105901742e+03_WP, -0.18917813819516756882830838328e+03_WP, 0.52780815920542364900561016686e+03_WP, -0.11573902539959630126141871134e+02_WP, 0.68812326946963000169666922661e+01_WP, -0.10006050966910838403183860980e+01_WP, 0.77771377980534432092869265740e+00_WP, -0.27782057523535084065932004339e+01_WP, -0.60196695231264120758267380846e+02_WP, 0.84320405506677161018159903784e+02_WP, 0.11992291136182789328035130030e+02_WP, -0.25693933462703749003312586129e+02_WP, -0.15418974869023643374053993627e+03_WP, -0.23152937917604549567536039109e+03_WP, 0.35763911791061412378285349910e+03_WP, 0.93405324183624310003907691704e+02_WP, -0.37458323136451633156875139351e+02_WP, 0.10409964950896230045147246184e+03_WP, 0.29840293426660503123344363579e+02_WP, -0.43533456590011143754432175058e+02_WP, 0.96324553959188282948394950600e+02_WP, -0.39177261675615439165231486172e+02_WP, -0.14972683625798562581422125276e+03_WP ], [12,4])
 dij, where i = 1 to s, j = 1 to pstar for a generic method
 
integer, parameter verner98r_q = 8
 min(p, phat) used in step-size computation
 
integer, parameter verner98r_p = 9
 Order of the propagating stage.
 
integer, parameter verner98r_phat = 8
 Order of the error stage.
 
integer, parameter verner98r_pstar = 8
 Interpolant order.
 
integer, parameter verner98r_s = 21
 Total Number of stages.
 
logical, parameter verner98r_fsal = .FALSE.
 Is this a FSAL method?
 
integer, parameter verner98r_sint = 16
 Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.
 
real(wp), dimension(2:*), parameter verner98r_c = [ 0.40e-1_WP, 0.9648736013787361245235039379666356743708e-1_WP, 0.1447310402068104186785255906949953511556_WP, 0.576_WP, 0.2272326564618766017153738192188229509142_WP, 0.5407673435381233982846261807811770490858_WP, 0.64_WP, 0.48_WP, 0.6754e-1_WP, 0.25_WP, 0.6770920153543242682384311058159603931192_WP, 0.8115_WP, 0.906_WP, 1.0_WP, 1.0_WP, 1.0_WP, 0.7421010083583088006243877166286162723673_WP, 0.888_WP, 0.696_WP, 0.487_WP ]
 c_i where i = 2 to s, s=number of stages
 
real(wp), dimension(*), parameter verner98r_b = [ 0.1458885278405539719101539582255752917034e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.2024197887889332650566666683195656097825e-2_WP, 0.2178047084569716646796256135839225745895_WP, 0.1274895340854389692868677968654808668201_WP, 0.2244617745463131861258531547137348031621_WP, 0.1787254491259903095100090833796054447157_WP, 0.7594344758096557172908303416513173076283e-1_WP, 0.1294845879197561516869001434704642286297_WP, 0.2947744761261941714007911131590716605202e-1_WP, 0.0_WP ]
 b_i where i = 1 to sint, sint=number of main stages These are higher order weights
 
real(wp), dimension(*), parameter verner98r_bh = [ 0.2034666655224434599707885098832906986649e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 1.069617650982700109541321983413338230042_WP, 0.7680834711303187278673130261850350530338e-1_WP, 0.1130778186885240437498706751119241126785_WP, 0.2552587357981962194892445789565762186511_WP, -0.9825898086919164036191607912120918904022_WP, 0.3981545824421514217762002137442675068982_WP, 0.0_WP, 0.0_WP, 0.4932600711506839027871318637915324696208e-1_WP ]
 bh_i where i = 1 to sint These are lower order weights
 
real(wp), dimension(*), parameter verner98r_e = Verner98R_b - Verner98R_bh
 e_i where i = 1 to sint
 
real(wp), dimension(1:*), parameter verner98r_a = [ .40e-1_WP, -.198852731918229097650241511466089129345e-1_WP, .1163726333296965222173745449432724803716_WP, .3618276005170260466963139767374883778890e-1_WP, 0.0_WP, .1085482801551078140088941930212465133667_WP, 2.272114264290177409193144938921415409241_WP, 0.0_WP, -8.526886447976398578316416192982602292786_WP, 6.830772183686221169123271254061186883545_WP, .5094385535389374394512668566783434123978e-1_WP, 0.0_WP, 0.0_WP, .1755865049809071110203693328749561646990_WP, .70229612707574674987780067603244497535e-3_WP, .1424783668683284782770955365543878809824_WP, 0.0_WP, 0.0_WP, -.3541799434668684104094753917518523845155_WP, .7595315450295100889001534202778550159932e-1_WP, .6765157656337123215269906939508560510196_WP, .7111111111111111111111111111111111111111e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3279909287605898328568406057725491803016_WP, .2408979601282990560320482831163397085872_WP, .7125e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3268842451575245554847578757216915662785_WP, .1156157548424754445152421242783084337215_WP, -.3375e-1_WP, .4822677322465810178387112087673611111111e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3948559980495400110769549704186108167677e-1_WP, .1058851161934658144373823566907778072121_WP, -0.2152006320474309346664428710937500000000e-1_WP, -.1045374260183348238623046875000000000000_WP, -.2609113435754923412210928689962011065179e-1_WP, 0.0_Wp, 0.0_WP, 0.0_Wp, 0.0_WP, .3333333333333333333333333333333333333333e-1_WP, -.1652504006638105086724681598195267241410_WP, .3434664118368616658319419895678838776647e-1_WP, .1595758283215209043195814910843067811951_WP, .2140857321828193385584684233447183324979_WP, -.362842339625565859076509979091267105528e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -1.096167597427208807028761474420297770752_WP, .1826035504321331052308236240517254331348_WP, .708225444417068325613028685455625123741e-1_WP, -.231364701848243126999929738482630407146e-1_WP, .2711204726320932916455631550463654973432_WP, 1.308133749422980744437146904349994472286_WP, -.5074635056416974879347823927726392374259_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -6.631342198657237090355284142048733580937_WP, -.252748010090880105270020973014860316405_WP, -.4952612380036095562991116175550167835424_WP, .293252554525388690285739720360003594753_WP, 1.440108693768280908474851998204423941413_WP, 6.237934498647055877243623886838802127716_WP, .7270192054526987638549835199880202544289_WP, .6130118256955931701496387847232542148725_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 9.088803891640463313341034206647776279557_WP, -.407378815629344868103315381138325162923_WP, 1.790733389490374687043894756399015035977_WP, .714927166761755073724875250629602731782_WP, -1.438580857841722850237810322456327208949_WP, -8.263329312064740580595954649844133476994_WP, -1.537570570808865115231450725068826856201_WP, .3453832827564871699090880801079644428793_WP, -1.211697910343873872490625222495537087293_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -19.05581871559595277753334676575234493500_WP, 1.26306067538987510135943101851905310045_WP, -6.913916969178458046793476128409110926069_WP, -.676462266509498065300115641383621209887_WP, 3.367860445026607887090352785684064242560_WP, 18.00675164312590810020103216906571965203_WP, 6.838828926794279896350389904990814350968_WP, -1.031516451921950498420447675652291096155_WP, .4129106232130622755368055554332539084021_WP, 2.157389007494053627033175177985666660692_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 23.80712219809580523172312179815279712750_WP, .88627792492165554903036801415266308369_WP, 13.13913039759876381480201677314222971522_WP, -2.604415709287714883747369630937415176632_WP, -5.193859949783872300189266203049579105962_WP, -20.41234071154150778768154893536134356354_WP, -12.30085625250572261314889445241581039623_WP, 1.521553095008539362178397458330791655267_WP, 0.0_WP, 0.0_WP, Verner98R_b(1:16), .1560140526108861651166696161636643678715e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .2681164393327584462067257298507705692190_WP, .1883053124587791093781990103116496723062_WP, .1249199137461030776325034278436425745187_WP, .2302302127814521984877267202569592992854_WP, -.1360312216132798635363504089171114829643_WP, .7488659971306952790649243500419796404442e-1_WP, -.2812840029795629049644800354887547182869e-1_WP, -.2314455726481949553795911074427483913356e-1_WP, 0.0_WP, .2734530424111347407183095495529155013292e-1_WP, .1311195721844068401556584238625026007821e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -.1464024265969826989710171144006401375029_WP, .2471264389666795959943268183777606168285_WP, .1311375203080032288323212687159488912255_WP, .2170560346982582529677346605154232633905_WP, .2867536713760320279685175119456430108611_WP, .2323311339149421811667180525022488323943e-1_WP, .5250677264199395668210993035986718117205e-1_WP, .2833951586009950740445688183974089075836e-2_WP, 0.0_WP, -.8502403851995712467765683685024083445736e-2_WP, .6914537026206649612108927235057202507752e-1_WP, .1398921213361768395172663025684236914687e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -.3157406517950500155656872013463001170279e-1_WP, .2271812513272158193884043729886141068401_WP, .1289486410996786560165007139967355200862_WP, .2216682589135277018738330104776507206003_WP, .1948368236542480784645408517950426740505_WP, .5740088404417652948912844658374513373184e-1_WP, .9008366542675954700472151691971957675586e-1_WP, .1579153208844212165580559397387558558087e-1_WP, 0.0_WP, -.1899131505909185814243914464042645862057e-1_WP, -.8830926811918835070775190595441414629725e-1_WP, -.1150256203298809274379013662627550701719_WP, .1615147291900762304051270768727488699116e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .8098685003242905691737114344809608208533e-1_WP, .1276916294306930449548063682447180310200_WP, .1234814359383480575498127887662911829596_WP, .2339851259140109824702007335541675692697_WP, -.6595995683357367659329999201079842018819e-1_WP, -.2565276859406432791053055820612070574951e-1_WP, -.1258973463819247076247947215937788425689_WP, -.4307672490364843615025132534342781501874e-1_WP, 0.0_WP, .4973042479196705159702504356466546825472e-1_WP, .1000473540179392626782905455570668634250_WP, .1378658806763623213069940499298119480365_WP, -.1223533770075462522361367835979662485167_WP ]
 a_ij, where i = 2 to s, j = 1 to i-1.
 
integer, dimension(*), parameter verner98r_di_nz = [1,(i, i=8,15),(i, i=17,21)]
 Coefficients for 8th-order interpolation.
 
real(wp), dimension(*, 1:*), parameter verner98r_d = reshape([ 1.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -12.75304069282388950483064356409920964903_WP, -.7205785602508598770412906345635211707530_WP, -48.06969107148755163304089843112677204750_WP, 16.32345788425353372538518290168630386345_WP, -5.888504109270884968456670963647316074790_WP, -69.22821100686856642029708151339949374410_WP, -38.04668072585188932845088326881300565231_WP, -75.21598899610186748511604166683735788867_WP, -19.46588639117053206710108537195779499764_WP, 22.25276964616901764060657108171977843291_WP, 14.38227638804283974976194859491865842966_WP, 94.92756297288050252130347529152607755873_WP, 63.97757128518312942674385099931772857860_WP, 57.52494337729701822053356654527592436144_WP, 68.54470113831162103032818060021729044674_WP, 6.559119452090996226782640921071801708575_WP, 451.8280048138745279924509263176669733181_WP, -118.7000544943430560099961188668917723161_WP, 89.44704113715942232261735606938508401319_WP, 627.4402883568152894700875088172124339413_WP, 340.9894379782233715580222226199659613703_WP, 670.5551756563966247587443735545005782764_WP, 172.8442419404515527792492527636420780662_WP, -202.2239493340537483960420170974810960961_WP, -301.8862921284913174289877784920239768093_WP, -819.3810521264963242304829331408227079441_WP, -398.4710369466142415586467341825347335632_WP, -587.5456254433247185141268798839079144120_WP, -194.8086610529652454702496599846926584629_WP, -23.65417183348355244612060745867113623926_WP, -1652.497181212881040972312045847339995538_WP, 379.7566858308294928207030358265089540700_WP, -380.5212519133325379643523316755978956588_WP, -2258.351433966898561277167615407280413114_WP, -1221.089637320158271327850635113537223709_WP, -2395.425253052829223230171636642290078449_WP, -616.3040486605512589130986770456245791880_WP, 741.1380586926791133828821084525361362850_WP, 1781.158815560130838385479471257488378388_WP, 2811.287710727701073955763185548552769132_WP, 716.5966869905804904541294272473572154569_WP, 2312.713681211178682602365980842590527027_WP, 317.7392440058917273091720330437842255568_WP, 44.16895609281556502164197054695369477469_WP, 3109.740640759030910750811309301753788556_WP, -658.7110535057816543655898314260197627370_WP, 770.9937272473377188110779497353201847420_WP, 4212.395800215491820633428188331388768232_WP, 2271.121533555259071203608357605468537656_WP, 4449.143231627101738352915330060972953361_WP, 1143.483637444009748019174885933341490497_WP, -1419.345991543901288934527221623894382397_WP, -4706.785555404445608825447884178551282728_WP, -4973.994583541372152306531194715881127387_WP, 52.89052166461839778508215175982407959882_WP, -4612.840108616055993454816044374461167725_WP, -299.5110317016553611442365899609354277447_WP, -45.39036357573745767095716350329312304563_WP, -3211.322751886536222454319056864879758831_WP, 644.2805206048469902599556429935393520064_WP, -831.4716893548379599332950159082545018560_WP, -4325.916149192490627782429268979382924699_WP, -2328.099120585713191256845645730711618239_WP, -4556.769050179329341174034103718438911683_WP, -1170.357669638643781845251579162353094100_WP, 1505.723453115174938364140876028053730124_WP, 6291.645831877559252216657958850293660213_WP, 4810.867980444981597632278117658725359609_WP, -1470.500844962700814722191656818947551329_WP, 4986.820885035081979510527485116584809575_WP, 151.9504248851541972968268022793780785647_WP, 24.43461389111572285100991773589739045465_WP, 1734.432866374712319698880133120955280438_WP, -334.8751825767893992539894957157903467478_WP, 461.9387177871916189774255799856083869291_WP, 2327.648261456242912420669452255092833474_WP, 1251.132631830924456968695609885535799254_WP, 2447.365350086656490540478347199404246857_WP, 628.2905498131297391288540900112436704439_WP, -841.3580375761078273196578619974820483141_WP, -4164.432457468105482852522949304279052327_WP, -2417.563170381452428143421579409061339218_WP, 1524.433133991196905505389714285761298379_WP, -2793.397702113869225818637760332264198189_WP, -32.14704772912899411981910701782974118232_WP, -5.395551268662524772664900940711910826177_WP, -383.8940830682559717177907419834455933213_WP, 72.05311579106953179281845208383275272780_WP, -104.2735790197010640588910140881002072916_WP, -513.8098304131662767347811744202515986459_WP, -275.9322212851025822454499429637433189491_WP, -539.5239805539746656111293686438409662456_WP, -138.4613470596128476846868080169758635556_WP, 193.8136970000397952625975451565478819654_WP, 1085.917381175309478755059233272153614833_WP, 493.8555519037577305710909287669609682492_WP, -488.9260320222638668905067532907780371207_WP, 636.7239265496922574541536520861820193631_WP ], [size(Verner98R_di_NZ),Verner98R_pstar])
 dij, where i = 1 to s, j = 1 to pstar for a generic method
 
integer, parameter verner65e_q = 5
 min(p, phat) used in step-size computation
 
integer, parameter verner65e_p = 6
 Order of the propagating stage.
 
integer, parameter verner65e_phat = 5
 Order of the error stage.
 
integer, parameter verner65e_pstar = 5
 Order of interpolants.
 
integer, parameter verner65e_s = 10
 Total Number of stages.
 
logical, parameter verner65e_fsal = .TRUE.
 Is this a FSAL method?
 
integer, parameter verner65e_sint = 9
 Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.
 
real(wp), dimension(2:*), parameter verner65e_c = [ 0.6e-1_WP, 0.9593333333333333333333333333333333333333e-1_WP, 0.1439_WP, 0.4973_WP, 0.9725_WP, 0.9995_WP, 1.0_WP, 1.0_WP, 0.5_WP ]
 c_i where i = 2 to s, s=number of stages
 
real(wp), dimension(*), parameter verner65e_b = [ 0.3438957868357036009278820124728322386520e-1_WP, 0.0_WP, 0.0_WP, 0.2582624555633503404659558098586120858767_WP, 0.4209371189673537150642551514069801967032_WP, 4.405396469669310170148836816197095664891_WP, -176.4831190242986576151740942499002125029_WP, 172.3641334014150730294022582711902413315_WP, 0.0_WP ]
 b_i where i = 1 to sint, sint=number of main stages These are higher order weights
 
real(wp), dimension(*), parameter verner65e_bh = [ .4909967648382489730906854927971225836479e-1_WP, 0.0_WP, 0.0_WP, 0.2251112229516524153401395320539875329485_WP, 0.4694682253029562039431948525047387412553_WP, 0.8065792249988867707634161808995217981443_WP, 0.0_WP, -0.6071194891777959797672951465256217122488_WP, 0.5686113944047569241147603178766138153594e-1_WP ]
 bh_i where i = 1 to sint These are lower order weights
 
real(wp), dimension(*), parameter verner65e_e = Verner65E_b - Verner65E_bh
 e_i where i = 1 to sint
 
real(wp), dimension(1:*), parameter verner65e_a = [ 0.6e-1_WP, .1923996296296296296296296296296296296296e-1_WP, .7669337037037037037037037037037037037037e-1_WP, .35975e-1_WP, 0.0_WP, .107925_WP, 1.318683415233148260919747276431735612861_WP, 0.0_WP, -5.042058063628562225427761634715637693344_WP, 4.220674648395413964508014358283902080483_WP, -41.87259166432751461803757780644346812905_WP, 0.0_WP, 159.4325621631374917700365669070346830453_WP, -122.1192135650100309202516203389242140663_WP, 5.531743066200053768252631238332999150076_WP, -54.43015693531650433250642051294142461271_WP, 0.0_WP, 207.0672513650184644273657173866509835987_WP, -158.6108137845899991828742424365058599469_WP, 6.991816585950242321992597280791793907096_WP, -.1859723106220323397765171799549294623692e-1_WP, -54.66374178728197680241215648050386959351_WP, 0.0_WP, 207.9528062553893734515824816699834244238_WP, -159.2889574744995071508959805871426654216_WP, 7.018743740796944434698170760964252490817_WP, -.1833878590504572306472782005141738268361e-1_WP, -.5119484997882099077875432497245168395840e-3_WP, Verner65E_b(1:8), 0.1652415901357280684383619367363197852645e-1_WP, 0.0_WP, 0.0_WP, 0.3053128187514178931377105638345583032476_WP, 0.2071200938201978848991082158995582390341_WP, -1.293879140655123187129665774327355723229_WP, 57.11988411588149149650780257779402737914_WP, -55.87979207510932290773937033203265749155_WP, 0.2483002829776601348057855515823731483430e-1_WP ]
 a_ij, where i = 2 to s, j = 1 to i-1.
 
integer, dimension(*), parameter verner65e_di_nz = [1,(i, i=4,10)]
 Coefficients for 5th-order interpolation.
 
real(wp), dimension(*, 1:*), parameter verner65e_d = reshape([ 1.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -5.308169607103576297743491917539437544903_WP, 6.272050253212501244827865529084399503479_WP, 6.876491702846304590450466371720363234704_WP, -35.54445171059960218765875699270358093032_WP, 1918.654856698011449175045220651610014945_WP, -1883.069802132718312960582779305006646556_WP, .1190247963512364356614756628348873476063_WP, -8.0_WP, 10.18168044895868030520877351032733768603_WP, -16.02618147467745958442607061022576892601_WP, -24.63576726084633318864583120149461053641_WP, 165.7016170190242105108446269288474846144_WP, -9268.121508966042500195164712930044037430_WP, 9101.025187200633795903395040749471730528_WP, -0.1250269670503937512118264468821359362429_WP, 32.0_WP, -7.520036991611714828300683961994073691563_WP, 12.84435632451961742214954703737612797249_WP, 33.21078648379717088772133447477731248517_WP, -385.4635395491142731464726480659809841649_WP, 20858.33702877255011893787944928058522511_WP, -20473.18855195953591834830509979123557878_WP, 1.779956919394999075328101026471971070697_WP, -40.0_WP, .9340485368631160925057442706475838478288_WP, -1.148794504476759027536609501260874665600_WP, -17.49461528263643828092150992351036511970_WP, 442.4324137015701845319394642134164121973_WP, -22645.82767158481047968149020787687967272_WP, 22209.76555125653413900516974418122400018_WP, -4.660932123043762639666625363637083723091_WP, 16.0_WP, .7468671915770650884224462998058729264688_WP, -1.683168143014549714548776645115271798480_WP, 2.464041475806649706459795429914280132942_WP, -182.7206429912112095385038492673822360516_WP, 8960.474176055992754148556156624828257597_WP, -8782.168250963498630570274647563263264047_WP, 2.886977374347920879888875121212361241030_WP, 0.0_WP ], [size(Verner65E_di_NZ),Verner65E_pstar])
 dij, where i = 1 to s, j = 1 to pstar for a generic method
 

Detailed Description

Butcher Tableau Module.

This module contains Butcher tableaus of each of the supported ERK methods in addition to specific parameters that are common to all the methods. Parameters that must be provided for each of the methods are: $ q $ is used in the error computation. Typically, $q = {min}(p, \hat{p})$. $ p $ is the order of the embedded Runge-Kutta method used to propagate the solution. $ \hat{p} $ is the order of the embedded Runge-Kutta method used to compute the error. The convention used for stages is that 'sint' represents the total number of stages needed for step integration including the FSAL stage for an FSAL method. So far, the coefficients for the following methods are included:

Author
Bharat Mahajan
Date
Created: 02/04/2019

Variable Documentation

◆ doi

integer butchertableaus::doi

Definition at line 74 of file ButcherTableaus.f90.

◆ dop54_a

real(wp), dimension(1:*), parameter butchertableaus::dop54_a = [ 1.0_WP/5.0_WP, 3.0_WP/40.0_WP, 9.0_WP/40.0_WP, 44.0_WP/45.0_WP, -56.0_WP/15.0_WP, 32.0_WP/9.0_WP, 19372.0_WP/6561.0_WP, -25360.0_WP/2187.0_WP, 64448.0_WP/6561.0_WP, -212.0_WP/729.0_WP, 9017.0_WP/3168.0_WP, -355.0_WP/33.0_WP, 46732.0_WP/5247.0_WP, 49.0_WP/176.0_WP, -5103.0_WP/18656.0_WP, DOP54_b(1:6) ]

a_ij, where i = 2 to s, j = 1 to i-1.

Remarks
Since a_ij is a lower triangular matrix, it is stored sequentially in a single dimension array for speedup. The storage sequence is a_21, a_31, a_32, a_41, a_42, a_43,...a_{s,s-1}.

Definition at line 145 of file ButcherTableaus.f90.

◆ dop54_b

real(wp), dimension(*), parameter butchertableaus::dop54_b = [ 35.0_WP/384.0_WP, 0.0_WP, 500.0_WP/1113.0_WP, 125.0_WP/192.0_WP, -2187.0_WP/6784.0_WP, 11.0_WP/84.0_WP, 0.0_WP ]

b_i where i = 1 to sint, sint=number of main stages These are higher order weights

Definition at line 112 of file ButcherTableaus.f90.

◆ dop54_beta

real(wp), parameter butchertableaus::dop54_beta = 0.04_WP

Lund stabilization parameters: beta and beta_multiplier. Positive values (< 0.1) make the step size control more stable. See Hairer's original DOPRI5 codes.

Definition at line 95 of file ButcherTableaus.f90.

◆ dop54_beta_mult

real(wp), parameter butchertableaus::dop54_beta_mult = 0.75_WP

Definition at line 96 of file ButcherTableaus.f90.

◆ dop54_bh

real(wp), dimension(*), parameter butchertableaus::dop54_bh = [ 5179.0_WP/57600.0_WP, 0.0_WP, 7571.0_WP/16695.0_WP, 393.0_WP/640.0_WP, -92097.0_WP/339200.0_WP, 187.0_WP/2100.0_WP, 1.0_WP/40.0_WP ]

bh_i where i = 1 to sint These are lower order weights for error estimation in local extrapolation case

Definition at line 126 of file ButcherTableaus.f90.

◆ dop54_c

real(wp), dimension(2:*), parameter butchertableaus::dop54_c = [ 1.0_WP/5.0_WP, 3.0_WP/10.0_WP, 4.0_WP/5.0_WP, 8.0_WP/9.0_WP, 1.0_WP, 1.0_WP ]

c_i where i = 2 to s, s=number of stages

Definition at line 99 of file ButcherTableaus.f90.

◆ dop54_d

real(wp), dimension(*,1:*), parameter butchertableaus::dop54_d = reshape([ -12715105075.0_WP/11282082432.0_WP, 87487479700.0_WP/32700410799.0_WP, -10690763975.0_WP/1880347072.0_WP, 701980252875.0_WP/199316789632.0_WP, -1453857185.0_WP/822651844.0_WP, 69997945.0_WP/29380423.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP ], [size(DOP54_di_NZ),2])

dij, where i = 1 to s, j = 1 to pstar for a generic method

Definition at line 176 of file ButcherTableaus.f90.

◆ dop54_di_nz

integer, dimension(*), parameter butchertableaus::dop54_di_nz = [1,3,4,5,6,7]

Coefficients for 4th-order interpolation.

Non-zero coefficients for non-zero stages

Definition at line 173 of file ButcherTableaus.f90.

◆ dop54_e

real(wp), dimension(*), parameter butchertableaus::dop54_e = DOP54_b - DOP54_bh

e_i where i = 1 to sint

Definition at line 139 of file ButcherTableaus.f90.

◆ dop54_fsal

logical, parameter butchertableaus::dop54_fsal = .TRUE.

Is this a FSAL method?

Definition at line 86 of file ButcherTableaus.f90.

◆ dop54_p

integer, parameter butchertableaus::dop54_p = 5

Order of the propagating stage.

Definition at line 82 of file ButcherTableaus.f90.

◆ dop54_phat

integer, parameter butchertableaus::dop54_phat = 4

Order of the error stage.

Definition at line 83 of file ButcherTableaus.f90.

◆ dop54_pstar

integer, parameter butchertableaus::dop54_pstar = 4

Order of interpolants.

Definition at line 84 of file ButcherTableaus.f90.

◆ dop54_q

integer, parameter butchertableaus::dop54_q = 4

min(p, phat) used in step-size computation

Definition at line 81 of file ButcherTableaus.f90.

◆ dop54_s

integer, parameter butchertableaus::dop54_s = 7

Total Number of stages.

Definition at line 85 of file ButcherTableaus.f90.

◆ dop54_sint

integer, parameter butchertableaus::dop54_sint = 7

Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.

Definition at line 91 of file ButcherTableaus.f90.

◆ dop853_a

real(wp), dimension(1:*), parameter butchertableaus::dop853_a = [ 5.26001519587677318785587544488e-2_WP, 1.97250569845378994544595329183e-2_WP, 5.91751709536136983633785987549e-2_WP, 2.95875854768068491816892993775e-2_WP, 0.0_WP, 8.87627564304205475450678981324e-2_WP, 2.41365134159266685502369798665e-1_WP, 0.0_WP, -8.84549479328286085344864962717e-1_WP, 9.24834003261792003115737966543e-1_WP, 3.7037037037037037037037037037e-2_WP, 0.0_WP, 0.0_WP, 1.70828608729473871279604482173e-1_WP, 1.25467687566822425016691814123e-1_WP, 3.7109375e-2_WP, 0.0_WP, 0.0_WP, 1.70252211019544039314978060272e-1_WP, 6.02165389804559606850219397283e-2_WP, -1.7578125e-2_WP, 3.70920001185047927108779319836e-2_WP, 0.0_WP, 0.0_WP, 1.70383925712239993810214054705e-1_WP, 1.07262030446373284651809199168e-1_WP, -1.53194377486244017527936158236e-2_WP, 8.27378916381402288758473766002e-3_WP, 6.24110958716075717114429577812e-1_WP, 0.0_WP, 0.0_WP, -3.36089262944694129406857109825e0_WP, -8.68219346841726006818189891453e-1_WP, 2.75920996994467083049415600797e1_WP, 2.01540675504778934086186788979e1_WP, -4.34898841810699588477366255144e1_WP, 4.77662536438264365890433908527e-1_WP, 0.0_WP, 0.0_WP, -2.48811461997166764192642586468e0_WP, -5.90290826836842996371446475743e-1_WP, 2.12300514481811942347288949897e1_WP, 1.52792336328824235832596922938e1_WP, -3.32882109689848629194453265587e1_WP, -2.03312017085086261358222928593e-2_WP, -9.3714243008598732571704021658e-1_WP, 0.0_Wp, 0.0_WP, 5.18637242884406370830023853209e0_WP, 1.09143734899672957818500254654e0_WP, -8.14978701074692612513997267357e0_WP, -1.85200656599969598641566180701e1_WP, 2.27394870993505042818970056734e1_WP, 2.49360555267965238987089396762e0_WP, -3.0467644718982195003823669022e0_WP, 2.27331014751653820792359768449e0_WP, 0.0_WP, 0.0_WP, -1.05344954667372501984066689879e1_WP, -2.00087205822486249909675718444e0_WP, -1.79589318631187989172765950534e1_WP, 2.79488845294199600508499808837e1_WP, -2.85899827713502369474065508674e0_WP, -8.87285693353062954433549289258e0_WP, 1.23605671757943030647266201528e1_WP, 6.43392746015763530355970484046e-1_WP, DOP853_b(1:12), 5.61675022830479523392909219681e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 2.53500210216624811088794765333e-1_WP, -2.46239037470802489917441475441e-1_WP, -1.24191423263816360469010140626e-1_WP, 1.5329179827876569731206322685e-1_WP, 8.20105229563468988491666602057e-3_WP, 7.56789766054569976138603589584e-3_WP, -8.298e-3_WP, 3.18346481635021405060768473261e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 2.83009096723667755288322961402e-2_WP, 5.35419883074385676223797384372e-2_WP, -5.49237485713909884646569340306e-2_WP, 0.0_WP, 0.0_WP, -1.08347328697249322858509316994e-4_WP, 3.82571090835658412954920192323e-4_WP, -3.40465008687404560802977114492e-4_WP, 1.41312443674632500278074618366e-1_WP, -4.28896301583791923408573538692e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -4.69762141536116384314449447206e0_WP, 7.68342119606259904184240953878e0_WP, 4.06898981839711007970213554331e0_WP, 3.56727187455281109270669543021e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, -1.39902416515901462129418009734e-3_WP, 2.9475147891527723389556272149e0_WP, -9.15095847217987001081870187138e0_WP ]

a_ij, where i = 2 to s, j = 1 to i-1.

Remarks
Since a_ij is a lower triangular matrix, it is stored sequentially in a single dimension array for speedup. The storage sequence is a_21, a_31, a_32, a_41, a_42, a_43,...a_{s,s-1}.

Definition at line 294 of file ButcherTableaus.f90.

◆ dop853_b

real(wp), dimension(*), parameter butchertableaus::dop853_b = [ 5.42937341165687622380535766363e-2_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 4.45031289275240888144113950566e0_WP, 1.89151789931450038304281599044e0_WP, -5.8012039600105847814672114227e0_WP, 3.1116436695781989440891606237e-1_WP, -1.52160949662516078556178806805e-1_WP, 2.01365400804030348374776537501e-1_WP, 4.47106157277725905176885569043e-2_WP, 0.0_WP ]

b_i where i = 1 to sint, sint=number of main stages

Definition at line 246 of file ButcherTableaus.f90.

◆ dop853_beta

real(wp), parameter butchertableaus::dop853_beta = 0.0_WP

Lund stabilization parameters: beta and beta_multiplier. Positive values (< 0.1) make the step size control more stable. See Hairer's original DOPRI5 codes.

Definition at line 221 of file ButcherTableaus.f90.

◆ dop853_beta_mult

real(wp), parameter butchertableaus::dop853_beta_mult = 0.2_WP

Definition at line 222 of file ButcherTableaus.f90.

◆ dop853_bhh

real(wp), dimension(*), parameter butchertableaus::dop853_bhh = [ 0.244094488188976377952755905512e+00_WP, 0.733846688281611857341361741547e+00_WP, 0.220588235294117647058823529412e-01_WP ]

bhh_i where i = 1 to 3 (DOP853 3rd order corrector for 5th-order error estimator)

Definition at line 264 of file ButcherTableaus.f90.

◆ dop853_c

real(wp), dimension(2:*), parameter butchertableaus::dop853_c = [ 0.526001519587677318785587544488e-01_WP, 0.789002279381515978178381316732e-01_WP, 0.118350341907227396726757197510e+00_WP, 0.281649658092772603273242802490e+00_WP, 0.333333333333333333333333333333e+00_WP, 0.25e+00_WP, 0.307692307692307692307692307692e+00_WP, 0.651282051282051282051282051282e+00_WP, 0.6e+00_WP, 0.857142857142857142857142857142e+00_WP, 1.0_WP, 1.0_WP, 0.1e+00_WP, 0.2e+00_WP, 0.777777777777777777777777777778e+00_WP ]

c_i where i = 2 to s, s=number of stages

Definition at line 225 of file ButcherTableaus.f90.

◆ dop853_d

real(wp), dimension(*,4:*), parameter butchertableaus::dop853_d = reshape([ -0.84289382761090128651353491142e+01_WP, 0.56671495351937776962531783590e+00_WP, -0.30689499459498916912797304727e+01_WP, 0.23846676565120698287728149680e+01_WP, 0.21170345824450282767155149946e+01_WP, -0.87139158377797299206789907490e+00_WP, 0.22404374302607882758541771650e+01_WP, 0.63157877876946881815570249290e+00_WP, -0.88990336451333310820698117400e-01_WP, 0.18148505520854727256656404962e+02_WP, -0.91946323924783554000451984436e+01_WP, -0.44360363875948939664310572000e+01_WP, 0.10427508642579134603413151009e+02_WP, 0.24228349177525818288430175319e+03_WP, 0.16520045171727028198505394887e+03_WP, -0.37454675472269020279518312152e+03_WP, -0.22113666853125306036270938578e+02_WP, 0.77334326684722638389603898808e+01_WP, -0.30674084731089398182061213626e+02_WP, -0.93321305264302278729567221706e+01_WP, 0.15697238121770843886131091075e+02_WP, -0.31139403219565177677282850411e+02_WP, -0.93529243588444783865713862664e+01_WP, 0.35816841486394083752465898540e+02_WP, 0.19985053242002433820987653617e+02_WP, -0.38703730874935176555105901742e+03_WP, -0.18917813819516756882830838328e+03_WP, 0.52780815920542364900561016686e+03_WP, -0.11573902539959630126141871134e+02_WP, 0.68812326946963000169666922661e+01_WP, -0.10006050966910838403183860980e+01_WP, 0.77771377980534432092869265740e+00_WP, -0.27782057523535084065932004339e+01_WP, -0.60196695231264120758267380846e+02_WP, 0.84320405506677161018159903784e+02_WP, 0.11992291136182789328035130030e+02_WP, -0.25693933462703749003312586129e+02_WP, -0.15418974869023643374053993627e+03_WP, -0.23152937917604549567536039109e+03_WP, 0.35763911791061412378285349910e+03_WP, 0.93405324183624310003907691704e+02_WP, -0.37458323136451633156875139351e+02_WP, 0.10409964950896230045147246184e+03_WP, 0.29840293426660503123344363579e+02_WP, -0.43533456590011143754432175058e+02_WP, 0.96324553959188282948394950600e+02_WP, -0.39177261675615439165231486172e+02_WP, -0.14972683625798562581422125276e+03_WP ], [12,4])

dij, where i = 1 to s, j = 1 to pstar for a generic method

Definition at line 433 of file ButcherTableaus.f90.

◆ dop853_di_nz

integer, dimension(*), parameter butchertableaus::dop853_di_nz = [ 1, (i, i=6,16)]

Non-zero coefficients for non-zero stages.

Definition at line 429 of file ButcherTableaus.f90.

◆ dop853_dj_nz

integer, dimension(*), parameter butchertableaus::dop853_dj_nz = [4,5,6,7]

Definition at line 430 of file ButcherTableaus.f90.

◆ dop853_e

real(wp), dimension(*), parameter butchertableaus::dop853_e = [ 0.1312004499419488073250102996e-01_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -0.1225156446376204440720569753e+01_WP, -0.4957589496572501915214079952e+00_WP, 0.1664377182454986536961530415e+01_WP, -0.3503288487499736816886487290e+00_WP, 0.3341791187130174790297318841e+00_WP, 0.8192320648511571246570742613e-01_WP, -0.2235530786388629525884427845e-01_WP, 0.0_WP ]

e_i where i = 1 to sint

Definition at line 272 of file ButcherTableaus.f90.

◆ dop853_fsal

logical, parameter butchertableaus::dop853_fsal = .TRUE.

Is this a FSAL method?

Definition at line 212 of file ButcherTableaus.f90.

◆ dop853_p

integer, parameter butchertableaus::dop853_p = 8

Order of the propagating stage.

Definition at line 208 of file ButcherTableaus.f90.

◆ dop853_phat

integer, parameter butchertableaus::dop853_phat = 6

Order of the error stage.

Definition at line 209 of file ButcherTableaus.f90.

◆ dop853_pstar

integer, parameter butchertableaus::dop853_pstar = 7

number of non-zero interpolation coefficients

Definition at line 210 of file ButcherTableaus.f90.

◆ dop853_q

integer, parameter butchertableaus::dop853_q = 7

min(p, phat) used in step-size computation

Definition at line 207 of file ButcherTableaus.f90.

◆ dop853_s

integer, parameter butchertableaus::dop853_s = 16

Total Number of stages.

Definition at line 211 of file ButcherTableaus.f90.

◆ dop853_sint

integer, parameter butchertableaus::dop853_sint = 13

Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.

Definition at line 217 of file ButcherTableaus.f90.

◆ erk_max_a

integer, parameter butchertableaus::erk_max_a = int((ERK_MAXSTAGES-1)/2.0*ERK_MAXSTAGES)

Definition at line 70 of file ButcherTableaus.f90.

◆ erk_maxipdegree

integer, parameter butchertableaus::erk_maxipdegree = 8

Definition at line 68 of file ButcherTableaus.f90.

◆ erk_maxstages

integer, parameter butchertableaus::erk_maxstages = 21

Maximum number of stages for storage allocation.

Definition at line 67 of file ButcherTableaus.f90.

◆ i

integer butchertableaus::i

Coefficients for 7th-order interpolation.

Definition at line 426 of file ButcherTableaus.f90.

◆ verner65e_a

real(wp), dimension(1:*), parameter butchertableaus::verner65e_a = [ 0.6e-1_WP, .1923996296296296296296296296296296296296e-1_WP, .7669337037037037037037037037037037037037e-1_WP, .35975e-1_WP, 0.0_WP, .107925_WP, 1.318683415233148260919747276431735612861_WP, 0.0_WP, -5.042058063628562225427761634715637693344_WP, 4.220674648395413964508014358283902080483_WP, -41.87259166432751461803757780644346812905_WP, 0.0_WP, 159.4325621631374917700365669070346830453_WP, -122.1192135650100309202516203389242140663_WP, 5.531743066200053768252631238332999150076_WP, -54.43015693531650433250642051294142461271_WP, 0.0_WP, 207.0672513650184644273657173866509835987_WP, -158.6108137845899991828742424365058599469_WP, 6.991816585950242321992597280791793907096_WP, -.1859723106220323397765171799549294623692e-1_WP, -54.66374178728197680241215648050386959351_WP, 0.0_WP, 207.9528062553893734515824816699834244238_WP, -159.2889574744995071508959805871426654216_WP, 7.018743740796944434698170760964252490817_WP, -.1833878590504572306472782005141738268361e-1_WP, -.5119484997882099077875432497245168395840e-3_WP, Verner65E_b(1:8), 0.1652415901357280684383619367363197852645e-1_WP, 0.0_WP, 0.0_WP, 0.3053128187514178931377105638345583032476_WP, 0.2071200938201978848991082158995582390341_WP, -1.293879140655123187129665774327355723229_WP, 57.11988411588149149650780257779402737914_WP, -55.87979207510932290773937033203265749155_WP, 0.2483002829776601348057855515823731483430e-1_WP ]

a_ij, where i = 2 to s, j = 1 to i-1.

Remarks
Since a_ij is a lower triangular matrix, it is stored sequentially in a single dimension array for speedup. The storage sequence is a_21, a_31, a_32, a_41, a_42, a_43,...a_{s,s-1}.

Definition at line 1038 of file ButcherTableaus.f90.

◆ verner65e_b

real(wp), dimension(*), parameter butchertableaus::verner65e_b = [ 0.3438957868357036009278820124728322386520e-1_WP, 0.0_WP, 0.0_WP, 0.2582624555633503404659558098586120858767_WP, 0.4209371189673537150642551514069801967032_WP, 4.405396469669310170148836816197095664891_WP, -176.4831190242986576151740942499002125029_WP, 172.3641334014150730294022582711902413315_WP, 0.0_WP ]

b_i where i = 1 to sint, sint=number of main stages These are higher order weights

Definition at line 1001 of file ButcherTableaus.f90.

◆ verner65e_bh

real(wp), dimension(*), parameter butchertableaus::verner65e_bh = [ .4909967648382489730906854927971225836479e-1_WP, 0.0_WP, 0.0_WP, 0.2251112229516524153401395320539875329485_WP, 0.4694682253029562039431948525047387412553_WP, 0.8065792249988867707634161808995217981443_WP, 0.0_WP, -0.6071194891777959797672951465256217122488_WP, 0.5686113944047569241147603178766138153594e-1_WP ]

bh_i where i = 1 to sint These are lower order weights

Definition at line 1017 of file ButcherTableaus.f90.

◆ verner65e_c

real(wp), dimension(2:*), parameter butchertableaus::verner65e_c = [ 0.6e-1_WP, 0.9593333333333333333333333333333333333333e-1_WP, 0.1439_WP, 0.4973_WP, 0.9725_WP, 0.9995_WP, 1.0_WP, 1.0_WP, 0.5_WP ]

c_i where i = 2 to s, s=number of stages

Definition at line 985 of file ButcherTableaus.f90.

◆ verner65e_d

real(wp), dimension(*,1:*), parameter butchertableaus::verner65e_d = reshape([ 1.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -5.308169607103576297743491917539437544903_WP, 6.272050253212501244827865529084399503479_WP, 6.876491702846304590450466371720363234704_WP, -35.54445171059960218765875699270358093032_WP, 1918.654856698011449175045220651610014945_WP, -1883.069802132718312960582779305006646556_WP, .1190247963512364356614756628348873476063_WP, -8.0_WP, 10.18168044895868030520877351032733768603_WP, -16.02618147467745958442607061022576892601_WP, -24.63576726084633318864583120149461053641_WP, 165.7016170190242105108446269288474846144_WP, -9268.121508966042500195164712930044037430_WP, 9101.025187200633795903395040749471730528_WP, -0.1250269670503937512118264468821359362429_WP, 32.0_WP, -7.520036991611714828300683961994073691563_WP, 12.84435632451961742214954703737612797249_WP, 33.21078648379717088772133447477731248517_WP, -385.4635395491142731464726480659809841649_WP, 20858.33702877255011893787944928058522511_WP, -20473.18855195953591834830509979123557878_WP, 1.779956919394999075328101026471971070697_WP, -40.0_WP, .9340485368631160925057442706475838478288_WP, -1.148794504476759027536609501260874665600_WP, -17.49461528263643828092150992351036511970_WP, 442.4324137015701845319394642134164121973_WP, -22645.82767158481047968149020787687967272_WP, 22209.76555125653413900516974418122400018_WP, -4.660932123043762639666625363637083723091_WP, 16.0_WP, .7468671915770650884224462998058729264688_WP, -1.683168143014549714548776645115271798480_WP, 2.464041475806649706459795429914280132942_WP, -182.7206429912112095385038492673822360516_WP, 8960.474176055992754148556156624828257597_WP, -8782.168250963498630570274647563263264047_WP, 2.886977374347920879888875121212361241030_WP, 0.0_WP ], [size(Verner65E_di_NZ),Verner65E_pstar])

dij, where i = 1 to s, j = 1 to pstar for a generic method

Definition at line 1095 of file ButcherTableaus.f90.

◆ verner65e_di_nz

integer, dimension(*), parameter butchertableaus::verner65e_di_nz = [1,(i, i=4,10)]

Coefficients for 5th-order interpolation.

Non-zero coefficients for non-zero stages

Definition at line 1092 of file ButcherTableaus.f90.

◆ verner65e_e

real(wp), dimension(*), parameter butchertableaus::verner65e_e = Verner65E_b - Verner65E_bh

e_i where i = 1 to sint

Definition at line 1032 of file ButcherTableaus.f90.

◆ verner65e_fsal

logical, parameter butchertableaus::verner65e_fsal = .TRUE.

Is this a FSAL method?

Definition at line 977 of file ButcherTableaus.f90.

◆ verner65e_p

integer, parameter butchertableaus::verner65e_p = 6

Order of the propagating stage.

Definition at line 973 of file ButcherTableaus.f90.

◆ verner65e_phat

integer, parameter butchertableaus::verner65e_phat = 5

Order of the error stage.

Definition at line 974 of file ButcherTableaus.f90.

◆ verner65e_pstar

integer, parameter butchertableaus::verner65e_pstar = 5

Order of interpolants.

Definition at line 975 of file ButcherTableaus.f90.

◆ verner65e_q

integer, parameter butchertableaus::verner65e_q = 5

min(p, phat) used in step-size computation

Definition at line 972 of file ButcherTableaus.f90.

◆ verner65e_s

integer, parameter butchertableaus::verner65e_s = 10

Total Number of stages.

Definition at line 976 of file ButcherTableaus.f90.

◆ verner65e_sint

integer, parameter butchertableaus::verner65e_sint = 9

Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.

Definition at line 982 of file ButcherTableaus.f90.

◆ verner98r_a

real(wp), dimension(1:*), parameter butchertableaus::verner98r_a = [ .40e-1_WP, -.198852731918229097650241511466089129345e-1_WP, .1163726333296965222173745449432724803716_WP, .3618276005170260466963139767374883778890e-1_WP, 0.0_WP, .1085482801551078140088941930212465133667_WP, 2.272114264290177409193144938921415409241_WP, 0.0_WP, -8.526886447976398578316416192982602292786_WP, 6.830772183686221169123271254061186883545_WP, .5094385535389374394512668566783434123978e-1_WP, 0.0_WP, 0.0_WP, .1755865049809071110203693328749561646990_WP, .70229612707574674987780067603244497535e-3_WP, .1424783668683284782770955365543878809824_WP, 0.0_WP, 0.0_WP, -.3541799434668684104094753917518523845155_WP, .7595315450295100889001534202778550159932e-1_WP, .6765157656337123215269906939508560510196_WP, .7111111111111111111111111111111111111111e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3279909287605898328568406057725491803016_WP, .2408979601282990560320482831163397085872_WP, .7125e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3268842451575245554847578757216915662785_WP, .1156157548424754445152421242783084337215_WP, -.3375e-1_WP, .4822677322465810178387112087673611111111e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .3948559980495400110769549704186108167677e-1_WP, .1058851161934658144373823566907778072121_WP, -0.2152006320474309346664428710937500000000e-1_WP, -.1045374260183348238623046875000000000000_WP, -.2609113435754923412210928689962011065179e-1_WP, 0.0_Wp, 0.0_WP, 0.0_Wp, 0.0_WP, .3333333333333333333333333333333333333333e-1_WP, -.1652504006638105086724681598195267241410_WP, .3434664118368616658319419895678838776647e-1_WP, .1595758283215209043195814910843067811951_WP, .2140857321828193385584684233447183324979_WP, -.362842339625565859076509979091267105528e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -1.096167597427208807028761474420297770752_WP, .1826035504321331052308236240517254331348_WP, .708225444417068325613028685455625123741e-1_WP, -.231364701848243126999929738482630407146e-1_WP, .2711204726320932916455631550463654973432_WP, 1.308133749422980744437146904349994472286_WP, -.5074635056416974879347823927726392374259_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -6.631342198657237090355284142048733580937_WP, -.252748010090880105270020973014860316405_WP, -.4952612380036095562991116175550167835424_WP, .293252554525388690285739720360003594753_WP, 1.440108693768280908474851998204423941413_WP, 6.237934498647055877243623886838802127716_WP, .7270192054526987638549835199880202544289_WP, .6130118256955931701496387847232542148725_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 9.088803891640463313341034206647776279557_WP, -.407378815629344868103315381138325162923_WP, 1.790733389490374687043894756399015035977_WP, .714927166761755073724875250629602731782_WP, -1.438580857841722850237810322456327208949_WP, -8.263329312064740580595954649844133476994_WP, -1.537570570808865115231450725068826856201_WP, .3453832827564871699090880801079644428793_WP, -1.211697910343873872490625222495537087293_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -19.05581871559595277753334676575234493500_WP, 1.26306067538987510135943101851905310045_WP, -6.913916969178458046793476128409110926069_WP, -.676462266509498065300115641383621209887_WP, 3.367860445026607887090352785684064242560_WP, 18.00675164312590810020103216906571965203_WP, 6.838828926794279896350389904990814350968_WP, -1.031516451921950498420447675652291096155_WP, .4129106232130622755368055554332539084021_WP, 2.157389007494053627033175177985666660692_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 23.80712219809580523172312179815279712750_WP, .88627792492165554903036801415266308369_WP, 13.13913039759876381480201677314222971522_WP, -2.604415709287714883747369630937415176632_WP, -5.193859949783872300189266203049579105962_WP, -20.41234071154150778768154893536134356354_WP, -12.30085625250572261314889445241581039623_WP, 1.521553095008539362178397458330791655267_WP, 0.0_WP, 0.0_WP, Verner98R_b(1:16), .1560140526108861651166696161636643678715e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .2681164393327584462067257298507705692190_WP, .1883053124587791093781990103116496723062_WP, .1249199137461030776325034278436425745187_WP, .2302302127814521984877267202569592992854_WP, -.1360312216132798635363504089171114829643_WP, .7488659971306952790649243500419796404442e-1_WP, -.2812840029795629049644800354887547182869e-1_WP, -.2314455726481949553795911074427483913356e-1_WP, 0.0_WP, .2734530424111347407183095495529155013292e-1_WP, .1311195721844068401556584238625026007821e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -.1464024265969826989710171144006401375029_WP, .2471264389666795959943268183777606168285_WP, .1311375203080032288323212687159488912255_WP, .2170560346982582529677346605154232633905_WP, .2867536713760320279685175119456430108611_WP, .2323311339149421811667180525022488323943e-1_WP, .5250677264199395668210993035986718117205e-1_WP, .2833951586009950740445688183974089075836e-2_WP, 0.0_WP, -.8502403851995712467765683685024083445736e-2_WP, .6914537026206649612108927235057202507752e-1_WP, .1398921213361768395172663025684236914687e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -.3157406517950500155656872013463001170279e-1_WP, .2271812513272158193884043729886141068401_WP, .1289486410996786560165007139967355200862_WP, .2216682589135277018738330104776507206003_WP, .1948368236542480784645408517950426740505_WP, .5740088404417652948912844658374513373184e-1_WP, .9008366542675954700472151691971957675586e-1_WP, .1579153208844212165580559397387558558087e-1_WP, 0.0_WP, -.1899131505909185814243914464042645862057e-1_WP, -.8830926811918835070775190595441414629725e-1_WP, -.1150256203298809274379013662627550701719_WP, .1615147291900762304051270768727488699116e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, .8098685003242905691737114344809608208533e-1_WP, .1276916294306930449548063682447180310200_WP, .1234814359383480575498127887662911829596_WP, .2339851259140109824702007335541675692697_WP, -.6595995683357367659329999201079842018819e-1_WP, -.2565276859406432791053055820612070574951e-1_WP, -.1258973463819247076247947215937788425689_WP, -.4307672490364843615025132534342781501874e-1_WP, 0.0_WP, .4973042479196705159702504356466546825472e-1_WP, .1000473540179392626782905455570668634250_WP, .1378658806763623213069940499298119480365_WP, -.1223533770075462522361367835979662485167_WP ]

a_ij, where i = 2 to s, j = 1 to i-1.

Remarks
Since a_ij is a lower triangular matrix, it is stored sequentially in a single dimension array for speedup. The storage sequence is a_21, a_31, a_32, a_41, a_42, a_43,...a_{s,s-1}.

Definition at line 610 of file ButcherTableaus.f90.

◆ verner98r_b

real(wp), dimension(*), parameter butchertableaus::verner98r_b = [ 0.1458885278405539719101539582255752917034e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.2024197887889332650566666683195656097825e-2_WP, 0.2178047084569716646796256135839225745895_WP, 0.1274895340854389692868677968654808668201_WP, 0.2244617745463131861258531547137348031621_WP, 0.1787254491259903095100090833796054447157_WP, 0.7594344758096557172908303416513173076283e-1_WP, 0.1294845879197561516869001434704642286297_WP, 0.2947744761261941714007911131590716605202e-1_WP, 0.0_WP ]

b_i where i = 1 to sint, sint=number of main stages These are higher order weights

Definition at line 559 of file ButcherTableaus.f90.

◆ verner98r_bh

real(wp), dimension(*), parameter butchertableaus::verner98r_bh = [ 0.2034666655224434599707885098832906986649e-1_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 1.069617650982700109541321983413338230042_WP, 0.7680834711303187278673130261850350530338e-1_WP, 0.1130778186885240437498706751119241126785_WP, 0.2552587357981962194892445789565762186511_WP, -0.9825898086919164036191607912120918904022_WP, 0.3981545824421514217762002137442675068982_WP, 0.0_WP, 0.0_WP, 0.4932600711506839027871318637915324696208e-1_WP ]

bh_i where i = 1 to sint These are lower order weights

Definition at line 582 of file ButcherTableaus.f90.

◆ verner98r_c

real(wp), dimension(2:*), parameter butchertableaus::verner98r_c = [ 0.40e-1_WP, 0.9648736013787361245235039379666356743708e-1_WP, 0.1447310402068104186785255906949953511556_WP, 0.576_WP, 0.2272326564618766017153738192188229509142_WP, 0.5407673435381233982846261807811770490858_WP, 0.64_WP, 0.48_WP, 0.6754e-1_WP, 0.25_WP, 0.6770920153543242682384311058159603931192_WP, 0.8115_WP, 0.906_WP, 1.0_WP, 1.0_WP, 1.0_WP, 0.7421010083583088006243877166286162723673_WP, 0.888_WP, 0.696_WP, 0.487_WP ]

c_i where i = 2 to s, s=number of stages

Definition at line 532 of file ButcherTableaus.f90.

◆ verner98r_d

real(wp), dimension(*,1:*), parameter butchertableaus::verner98r_d = reshape([ 1.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, 0.0_WP, -12.75304069282388950483064356409920964903_WP, -.7205785602508598770412906345635211707530_WP, -48.06969107148755163304089843112677204750_WP, 16.32345788425353372538518290168630386345_WP, -5.888504109270884968456670963647316074790_WP, -69.22821100686856642029708151339949374410_WP, -38.04668072585188932845088326881300565231_WP, -75.21598899610186748511604166683735788867_WP, -19.46588639117053206710108537195779499764_WP, 22.25276964616901764060657108171977843291_WP, 14.38227638804283974976194859491865842966_WP, 94.92756297288050252130347529152607755873_WP, 63.97757128518312942674385099931772857860_WP, 57.52494337729701822053356654527592436144_WP, 68.54470113831162103032818060021729044674_WP, 6.559119452090996226782640921071801708575_WP, 451.8280048138745279924509263176669733181_WP, -118.7000544943430560099961188668917723161_WP, 89.44704113715942232261735606938508401319_WP, 627.4402883568152894700875088172124339413_WP, 340.9894379782233715580222226199659613703_WP, 670.5551756563966247587443735545005782764_WP, 172.8442419404515527792492527636420780662_WP, -202.2239493340537483960420170974810960961_WP, -301.8862921284913174289877784920239768093_WP, -819.3810521264963242304829331408227079441_WP, -398.4710369466142415586467341825347335632_WP, -587.5456254433247185141268798839079144120_WP, -194.8086610529652454702496599846926584629_WP, -23.65417183348355244612060745867113623926_WP, -1652.497181212881040972312045847339995538_WP, 379.7566858308294928207030358265089540700_WP, -380.5212519133325379643523316755978956588_WP, -2258.351433966898561277167615407280413114_WP, -1221.089637320158271327850635113537223709_WP, -2395.425253052829223230171636642290078449_WP, -616.3040486605512589130986770456245791880_WP, 741.1380586926791133828821084525361362850_WP, 1781.158815560130838385479471257488378388_WP, 2811.287710727701073955763185548552769132_WP, 716.5966869905804904541294272473572154569_WP, 2312.713681211178682602365980842590527027_WP, 317.7392440058917273091720330437842255568_WP, 44.16895609281556502164197054695369477469_WP, 3109.740640759030910750811309301753788556_WP, -658.7110535057816543655898314260197627370_WP, 770.9937272473377188110779497353201847420_WP, 4212.395800215491820633428188331388768232_WP, 2271.121533555259071203608357605468537656_WP, 4449.143231627101738352915330060972953361_WP, 1143.483637444009748019174885933341490497_WP, -1419.345991543901288934527221623894382397_WP, -4706.785555404445608825447884178551282728_WP, -4973.994583541372152306531194715881127387_WP, 52.89052166461839778508215175982407959882_WP, -4612.840108616055993454816044374461167725_WP, -299.5110317016553611442365899609354277447_WP, -45.39036357573745767095716350329312304563_WP, -3211.322751886536222454319056864879758831_WP, 644.2805206048469902599556429935393520064_WP, -831.4716893548379599332950159082545018560_WP, -4325.916149192490627782429268979382924699_WP, -2328.099120585713191256845645730711618239_WP, -4556.769050179329341174034103718438911683_WP, -1170.357669638643781845251579162353094100_WP, 1505.723453115174938364140876028053730124_WP, 6291.645831877559252216657958850293660213_WP, 4810.867980444981597632278117658725359609_WP, -1470.500844962700814722191656818947551329_WP, 4986.820885035081979510527485116584809575_WP, 151.9504248851541972968268022793780785647_WP, 24.43461389111572285100991773589739045465_WP, 1734.432866374712319698880133120955280438_WP, -334.8751825767893992539894957157903467478_WP, 461.9387177871916189774255799856083869291_WP, 2327.648261456242912420669452255092833474_WP, 1251.132631830924456968695609885535799254_WP, 2447.365350086656490540478347199404246857_WP, 628.2905498131297391288540900112436704439_WP, -841.3580375761078273196578619974820483141_WP, -4164.432457468105482852522949304279052327_WP, -2417.563170381452428143421579409061339218_WP, 1524.433133991196905505389714285761298379_WP, -2793.397702113869225818637760332264198189_WP, -32.14704772912899411981910701782974118232_WP, -5.395551268662524772664900940711910826177_WP, -383.8940830682559717177907419834455933213_WP, 72.05311579106953179281845208383275272780_WP, -104.2735790197010640588910140881002072916_WP, -513.8098304131662767347811744202515986459_WP, -275.9322212851025822454499429637433189491_WP, -539.5239805539746656111293686438409662456_WP, -138.4613470596128476846868080169758635556_WP, 193.8136970000397952625975451565478819654_WP, 1085.917381175309478755059233272153614833_WP, 493.8555519037577305710909287669609682492_WP, -488.9260320222638668905067532907780371207_WP, 636.7239265496922574541536520861820193631_WP ], [size(Verner98R_di_NZ),Verner98R_pstar])

dij, where i = 1 to s, j = 1 to pstar for a generic method

Definition at line 835 of file ButcherTableaus.f90.

◆ verner98r_di_nz

integer, dimension(*), parameter butchertableaus::verner98r_di_nz = [1,(i, i=8,15),(i, i=17,21)]

Coefficients for 8th-order interpolation.

Non-zero coefficients for non-zero stages

Definition at line 832 of file ButcherTableaus.f90.

◆ verner98r_e

real(wp), dimension(*), parameter butchertableaus::verner98r_e = Verner98R_b - Verner98R_bh

e_i where i = 1 to sint

Definition at line 604 of file ButcherTableaus.f90.

◆ verner98r_fsal

logical, parameter butchertableaus::verner98r_fsal = .FALSE.

Is this a FSAL method?

Definition at line 524 of file ButcherTableaus.f90.

◆ verner98r_p

integer, parameter butchertableaus::verner98r_p = 9

Order of the propagating stage.

Definition at line 520 of file ButcherTableaus.f90.

◆ verner98r_phat

integer, parameter butchertableaus::verner98r_phat = 8

Order of the error stage.

Definition at line 521 of file ButcherTableaus.f90.

◆ verner98r_pstar

integer, parameter butchertableaus::verner98r_pstar = 8

Interpolant order.

Definition at line 522 of file ButcherTableaus.f90.

◆ verner98r_q

integer, parameter butchertableaus::verner98r_q = 8

min(p, phat) used in step-size computation

Definition at line 519 of file ButcherTableaus.f90.

◆ verner98r_s

integer, parameter butchertableaus::verner98r_s = 21

Total Number of stages.

Definition at line 523 of file ButcherTableaus.f90.

◆ verner98r_sint

integer, parameter butchertableaus::verner98r_sint = 16

Number of stages needed for integration without interpolation for dense output. It is assumed that these stages start from 2 to sint and the extra stages for interpolation occur after the integration stages.

Definition at line 529 of file ButcherTableaus.f90.