doubleexpintegrationconstants.hh

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#ifndef DUMUX_COMMON_DOUBLEEXP_INTEGRATION_CONSTANTS_HH
#define DUMUX_COMMON_DOUBLEEXP_INTEGRATION_CONSTANTS_HH
 
/*
The double exponential rule is based on the observation that the trapezoid rule converges
very rapidly for functions on the entire real line that go to zero like exp( - exp(t) ).
The change of variables x = tanh( pi sinh(t) /2) transforms an integral over [-1, 1]
into an integral with integrand suited to the double exponential rule.
 
The transformed integral is infinite, but we truncate the domain of integration to [-3, 3].
The limit '3' was chosen for two reasons: for t = 3, the transformed x values
are nearly equal to 1 for 12 or more significant figures.  Also, for t = 3, the
smallest weights are 12 orders of magnitude smaller than the largest weights; setting
the cutoff larger than 3 would not have a significant impact on the integral value
unless there is a strong singularity at one of the end points.
 
The change of variables x(t) is an odd function, i.e. x(-t) = -x(t), and so we need only
store the positive x values.  Also, the derivative w(t) = x'(t) is even, i.e. w(-t) = w(t),
and so we need only store the weights corresponding to positive values of x.
 
The integration first applies the trapezoid rule to [-3, 3] in steps of size 1.
Then it subsequently cuts the step size in half each time, comparing the results.
Integration stops when subsequent iterations are close enough together or the maximum
integration points have been used.
By cutting h in half, the previous integral can be reused; we only need evaluate the
integrand at the newly added points.
 
Finally, note that we're not strictly using the trapezoid rule: we don't treat the
end points differently.  This is because we assume the values at the ends of the interval
hardly matter due to the rapid decay of the integrand.
 
All values below were calculated with Mathematica.
*/
 
namespace Dumux {
 
constexpr double doubleExponentialIntegrationAbcissas[] =
{
    // 1st layer abcissas: transformed 0, 1, 2, 3
    0.00000000000000000000,
    0.95136796407274694573,
    0.99997747719246159286,
    0.99999999999995705839,
    // 2nd layer abcissas: transformed 1/2, 3/2, 5/2
    0.67427149224843582608,
    0.99751485645722438683,
    0.99999998887566488198,
    // 3rd layer abcissas: transformed 1/4, 3/4, ...
    0.37720973816403417379,
    0.85956905868989663517,
    0.98704056050737689169,
    0.99968826402835320905,
    0.99999920473711471266,
    0.99999999995285644818,
    // 4th layer abcissas: transformed 1/8, 3/8, ...
    0.19435700332493543161,
    0.53914670538796776905,
    0.78060743898320029925,
    0.91487926326457461091,
    0.97396686819567744856,
    0.99405550663140214329,
    0.99906519645578584642,
    0.99990938469514399984,
    0.99999531604122052843,
    0.99999989278161241838,
    0.99999999914270509218,
    0.99999999999823216531,
    // 5th layer abcissa: transformed 1/16, 3/16, ...
    0.097923885287832333262,
    0.28787993274271591456,
    0.46125354393958570440,
    0.61027365750063894488,
    0.73101803479256151149,
    0.82331700550640237006,
    0.88989140278426019808,
    0.93516085752198468323,
    0.96411216422354729193,
    0.98145482667733517003,
    0.99112699244169880223,
    0.99610866543750854254,
    0.99845420876769773751,
    0.99945143443527460584,
    0.99982882207287494166,
    0.99995387100562796075,
    0.99998948201481850361,
    0.99999801714059543208,
    0.99999969889415261122,
    0.99999996423908091534,
    0.99999999678719909830,
    0.99999999978973286224,
    0.99999999999039393352,
    0.99999999999970809734,
    // 6th layer abcissas: transformed 1/32, 3/32, ...
    0.049055967305077886315,
    0.14641798429058794053,
    0.24156631953888365838,
    0.33314226457763809244,
    0.41995211127844715849,
    0.50101338937930910152,
    0.57558449063515165995,
    0.64317675898520470128,
    0.70355000514714201566,
    0.75669390863372994941,
    0.80279874134324126576,
    0.84221924635075686382,
    0.87543539763040867837,
    0.90301328151357387064,
    0.92556863406861266645,
    0.94373478605275715685,
    0.95813602271021369012,
    0.96936673289691733517,
    0.97797623518666497298,
    0.98445883116743083087,
    0.98924843109013389601,
    0.99271699719682728538,
    0.99517602615532735426,
    0.99688031812819187372,
    0.99803333631543375402,
    0.99879353429880589929,
    0.99928111192179195541,
    0.99958475035151758732,
    0.99976797159956083506,
    0.99987486504878034648,
    0.99993501992508242369,
    0.99996759306794345976,
    0.99998451990227082442,
    0.99999293787666288565,
    0.99999693244919035751,
    0.99999873547186590954,
    0.99999950700571943689,
    0.99999981889371276701,
    0.99999993755407837378,
    0.99999997987450320175,
    0.99999999396413420165,
    0.99999999832336194826,
    0.99999999957078777261,
    0.99999999989927772326,
    0.99999999997845533741,
    0.99999999999582460688,
    0.99999999999927152627,
    0.99999999999988636130,
    // 7th layer abcissas: transformed 1/64, 3/64, ...
    0.024539763574649160379,
    0.073525122985671294475,
    0.12222912220155764235,
    0.17046797238201051811,
    0.21806347346971200463,
    0.26484507658344795046,
    0.31065178055284596083,
    0.35533382516507453330,
    0.39875415046723775644,
    0.44078959903390086627,
    0.48133184611690504422,
    0.52028805069123015958,
    0.55758122826077823080,
    0.59315035359195315880,
    0.62695020805104287950,
    0.65895099174335012438,
    0.68913772506166767176,
    0.71750946748732412721,
    0.74407838354734739913,
    0.76886868676824658459,
    0.79191549237614211447,
    0.81326360850297385168,
    0.83296629391941087564,
    0.85108400798784873261,
    0.86768317577564598669,
    0.88283498824466895513,
    0.89661425428007602579,
    0.90909831816302043511,
    0.92036605303195280235,
    0.93049693799715340631,
    0.93957022393327475539,
    0.94766419061515309734,
    0.95485549580502268541,
    0.96121861515111640753,
    0.96682537031235585284,
    0.97174454156548730892,
    0.97604156025657673933,
    0.97977827580061576265,
    0.98301279148110110558,
    0.98579936302528343597,
    0.98818835380074264243,
    0.99022624046752774694,
    0.99195566300267761562,
    0.99341551316926403900,
    0.99464105571251119672,
    0.99566407681695316965,
    0.99651305464025377317,
    0.99721334704346870224,
    0.99778739195890653083,
    0.99825491617199629344,
    0.99863314864067747762,
    0.99893703483351217373,
    0.99917944893488591716,
    0.99937140114093768690,
    0.99952223765121720422,
    0.99963983134560036519,
    0.99973076151980848263,
    0.99980048143113838630,
    0.99985347277311141171,
    0.99989338654759256426,
    0.99992317012928932869,
    0.99994518061445869309,
    0.99996128480785666613,
    0.99997294642523223656,
    0.99998130127012072679,
    0.99998722128200062811,
    0.99999136844834487344,
    0.99999423962761663478,
    0.99999620334716617675,
    0.99999752962380516793,
    0.99999841381096473542,
    0.99999899541068996962,
    0.99999937270733536947,
    0.99999961398855024275,
    0.99999976602333243312,
    0.99999986037121459941,
    0.99999991800479471056,
    0.99999995264266446185,
    0.99999997311323594362,
    0.99999998500307631173,
    0.99999999178645609907,
    0.99999999558563361584,
    0.99999999767323673790,
    0.99999999879798350040,
    0.99999999939177687583,
    0.99999999969875436925,
    0.99999999985405611550,
    0.99999999993088839501,
    0.99999999996803321674,
    0.99999999998556879008,
    0.99999999999364632387,
    0.99999999999727404948,
    0.99999999999886126543,
    0.99999999999953722654,
    0.99999999999981720098,
    0.99999999999992987953
}; // end abcissas
 
constexpr double doubleExponentialIntegrationWeights[] =
{
    // First layer weights
    1.5707963267948966192,
    0.230022394514788685,
    0.00026620051375271690866,
    1.3581784274539090834e-12,
    // 2nd layer weights
    0.96597657941230114801,
    0.018343166989927842087,
    2.1431204556943039358e-7,
    // 3rd layer weights
    1.3896147592472563229,
    0.53107827542805397476,
    0.076385743570832304188,
    0.0029025177479013135936,
    0.000011983701363170720047,
    1.1631165814255782766e-9,
    // 4th layer weights
    1.5232837186347052132,
    1.1934630258491569639,
    0.73743784836154784136,
    0.36046141846934367417,
    0.13742210773316772341,
    0.039175005493600779072,
    0.0077426010260642407123,
    0.00094994680428346871691,
    0.000062482559240744082891,
    1.8263320593710659699e-6,
    1.8687282268736410132e-8,
    4.9378538776631926964e-11,
    //  5th layer weights
    1.5587733555333301451,
    1.466014426716965781,
    1.297475750424977998,
    1.0816349854900704074,
    0.85017285645662006895,
    0.63040513516474369106,
    0.44083323627385823707,
    0.290240679312454185,
    0.17932441211072829296,
    0.10343215422333290062,
    0.055289683742240583845,
    0.027133510013712003219,
    0.012083543599157953493,
    0.0048162981439284630173,
    0.0016908739981426396472,
    0.00051339382406790336017,
    0.00013205234125609974879,
    0.000028110164327940134749,
    4.8237182032615502124e-6,
    6.4777566035929719908e-7,
    6.5835185127183396672e-8,
    4.8760060974240625869e-9,
    2.5216347918530148572e-10,
    8.6759314149796046502e-12,
    // 6th layer weights
    1.5677814313072218572,
    1.5438811161769592204,
    1.4972262225410362896,
    1.4300083548722996676,
    1.3452788847662516615,
    1.2467012074518577048,
    1.1382722433763053734,
    1.0240449331118114483,
    0.90787937915489531693,
    0.79324270082051671787,
    0.68306851634426375464,
    0.57967810308778764708,
    0.48475809121475539287,
    0.39938474152571713515,
    0.32408253961152890402,
    0.258904639514053516,
    0.20352399885860174519,
    0.15732620348436615027,
    0.11949741128869592428,
    0.089103139240941462841,
    0.065155533432536205042,
    0.046668208054846613644,
    0.032698732726609031113,
    0.022379471063648476483,
    0.014937835096050129696,
    0.0097072237393916892692,
    0.0061300376320830301252,
    0.0037542509774318343023,
    0.0022250827064786427022,
    0.0012733279447082382027,
    0.0007018595156842422708,
    0.00037166693621677760301,
    0.00018856442976700318572,
    0.000091390817490710122732,
    0.000042183183841757600604,
    0.000018481813599879217116,
    7.6595758525203162562e-6,
    2.9916615878138787094e-6,
    1.0968835125901264732e-6,
    3.7595411862360630091e-7,
    1.1992442782902770219e-7,
    3.5434777171421953043e-8,
    9.6498888961089633609e-9,
    2.4091773256475940779e-9,
    5.482835779709497755e-10,
    1.1306055347494680536e-10,
    2.0989335404511469109e-11,
    3.4841937670261059685e-12,
    // 7th layer weights
    1.5700420292795931467,
    1.5640214037732320999,
    1.5520531698454121192,
    1.5342817381543034316,
    1.5109197230741697127,
    1.48224329788553807,
    1.4485862549613225916,
    1.4103329714462590129,
    1.3679105116808964881,
    1.3217801174437728579,
    1.2724283455378627082,
    1.2203581095793582207,
    1.1660798699324345766,
    1.1101031939653403796,
    1.0529288799552666556,
    0.99504180404613271514,
    0.93690461274566793366,
    0.87895234555278212039,
    0.82158803526696470334,
    0.7651792989089561367,
    0.71005590120546898385,
    0.65650824613162753076,
    0.60478673057840362158,
    0.55510187800363350959,
    0.5076251588319080997,
    0.4624903980553677613,
    0.41979566844501548066,
    0.37960556938665160999,
    0.3419537959230168323,
    0.30684590941791694932,
    0.27426222968906810637,
    0.24416077786983990868,
    0.21648020911729617038,
    0.19114268413342749532,
    0.16805663794826916233,
    0.14711941325785693248,
    0.12821973363120098675,
    0.11123999898874453035,
    0.096058391865189467849,
    0.082550788110701737654,
    0.070592469906866999352,
    0.060059642358636300319,
    0.05083075757257047107,
    0.042787652157725676034,
    0.035816505604196436523,
    0.029808628117310126969,
    0.024661087314753282511,
    0.020277183817500123926,
    0.016566786254247575375,
    0.013446536605285730674,
    0.010839937168255907211,
    0.0086773307495391815854,
    0.0068957859690660035329,
    0.0054388997976239984331,
    0.0042565295990178580165,
    0.0033044669940348302363,
    0.0025440657675291729678,
    0.0019418357759843675814,
    0.0014690143599429791058,
    0.0011011261134519383862,
    0.00081754101332469493115,
    0.00060103987991147422573,
    0.00043739495615911687786,
    0.00031497209186021200274,
    0.00022435965205008549104,
    0.00015802788400701191949,
    0.00011002112846666697224,
    0.000075683996586201477788,
    0.000051421497447658802092,
    0.0000344921247593431977,
    0.000022832118109036146591,
    0.000014908514031870608449,
    9.5981941283784710776e-6,
    6.0899100320949039256e-6,
    3.8061983264644899045e-6,
    2.3421667208528096843e-6,
    1.4183067155493917523e-6,
    8.4473756384859863469e-7,
    4.9458288702754198508e-7,
    2.8449923659159806339e-7,
    1.6069394579076224911e-7,
    8.9071395140242387124e-8,
    4.8420950198072369669e-8,
    2.579956822953589238e-8,
    1.3464645522302038796e-8,
    6.8784610955899001111e-9,
    3.4371856744650090511e-9,
    1.6788897682161906807e-9,
    8.0099784479729665356e-10,
    3.7299501843052790038e-10,
    1.6939457789411646876e-10,
    7.4967397573818224522e-11,
    3.230446433325236576e-11,
    1.3542512912336274432e-11,
    5.5182369468174885821e-12,
    2.1835922099233609052e-12
}; // end weights
 
} // end namespace Dumux
 
#endif