G4DormandPrince745.cc

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//
// G4DormandPrince745 implementation
//
// DormandPrince7 - 5(4) non-FSAL
// definition of the stepper() method that evaluates one step in
// field propagation.
// The coefficients and the algorithm have been adapted from
//
// J. R. Dormand and P. J. Prince, "A family of embedded Runge-Kutta formulae"
// Journal of computational and applied Math., vol.6, no.1, pp.19-26, 1980.
//
// The Butcher table of the Dormand-Prince-7-4-5 method is as follows :
//
//    0   |
//    1/5 | 1/5
//    3/10| 3/40       9/40
//    4/5 | 44/45      56/15      32/9
//    8/9 | 19372/6561 25360/2187 64448/6561  212/729
//    1   | 9017/3168  355/33     46732/5247  49/176  5103/18656
//    1   | 35/384     0          500/1113    125/192 2187/6784    11/84
//    ------------------------------------------------------------------------
//          35/384     0          500/1113    125/192 2187/6784    11/84    0
//          5179/57600 0          7571/16695  393/640 92097/339200 187/2100 1/40
//
// Created: Somnath Banerjee, Google Summer of Code 2015, 25 May 2015
// Supervision: John Apostolakis, CERN
// --------------------------------------------------------------------

#include "G4DormandPrince745.hh"
#include "G4LineSection.hh"

#include <cstring>

using namespace field_utils;

G4DormandPrince745::G4DormandPrince745(G4EquationOfMotion* equation,
                                       G4int noIntegrationVariables)
    : G4MagIntegratorStepper(equation, noIntegrationVariables)
{
}

void G4DormandPrince745::Stepper(const G4double yInput[],
                                 const G4double dydx[],
                                       G4double hstep,
                                       G4double yOutput[],
                                       G4double yError[],
                                       G4double dydxOutput[])
{
  Stepper(yInput, dydx, hstep, yOutput, yError);
  copy(dydxOutput, ak7);
}

// Stepper
//
// Passing in the value of yInput[],the first time dydx[] and Step length
// Giving back yOut and yErr arrays for output and error respectively
//
void G4DormandPrince745::Stepper(const G4double yInput[],
                                 const G4double dydx[],
                                       G4double hstep,
                                       G4double yOut[],
                                       G4double yErr[])
{
    // The various constants defined on the basis of butcher tableu
    //
    const G4double b21 = 0.2,
                   b31 = 3.0 / 40.0, b32 = 9.0 / 40.0,
                   b41 = 44.0 / 45.0, b42 = -56.0 / 15.0, b43 = 32.0/9.0,

        b51 = 19372.0 / 6561.0, b52 = -25360.0 / 2187.0, b53 = 64448.0 / 6561.0,
        b54 = -212.0 / 729.0,
        
        b61 = 9017.0 / 3168.0 , b62 =   -355.0 / 33.0,
        b63 = 46732.0 / 5247.0, b64 = 49.0 / 176.0,
        b65 = -5103.0 / 18656.0,
        
        b71 = 35.0 / 384.0, b72 = 0.,
        b73 = 500.0 / 1113.0, b74 = 125.0 / 192.0,
        b75 = -2187.0 / 6784.0, b76 = 11.0 / 84.0,
    
    //Sum of columns, sum(bij) = ei
    //    e1 = 0. ,
    //    e2 = 1.0/5.0 ,
    //    e3 = 3.0/10.0 ,
    //    e4 = 4.0/5.0 ,
    //    e5 = 8.0/9.0 ,
    //    e6 = 1.0 ,
    //    e7 = 1.0 ,
    
    // Difference between the higher and the lower order method coeff. :
    // b7j are the coefficients of higher order
    
        dc1 = -(b71 - 5179.0 / 57600.0),
        dc2 = -(b72 - .0),
        dc3 = -(b73 - 7571.0 / 16695.0),
        dc4 = -(b74 - 393.0 / 640.0),
        dc5 = -(b75 + 92097.0 / 339200.0),
        dc6 = -(b76 - 187.0 / 2100.0),
        dc7 = -(- 1.0 / 40.0);
    
    const G4int numberOfVariables = GetNumberOfVariables();
    State yTemp;
    
    // The number of variables to be integrated over
    //
    yOut[7] = yTemp[7]  = yInput[7];

    //  Saving yInput because yInput and yOut can be aliases for same array
    //
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        fyIn[i] = yInput[i];
    }
    // RightHandSide(yIn, dydx);    // Not done! 1st stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + b21 * hstep * dydx[i];
    }
    RightHandSide(yTemp, ak2);              // 2nd stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + hstep * (b31 * dydx[i] + b32 * ak2[i]);
    }
    RightHandSide(yTemp, ak3);              // 3rd stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + hstep * (
            b41 * dydx[i] + b42 * ak2[i] + b43 * ak3[i]);
    }
    RightHandSide(yTemp, ak4);              // 4th stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + hstep * (
            b51 * dydx[i] + b52 * ak2[i] + b53 * ak3[i] + b54 * ak4[i]);
    }
    RightHandSide(yTemp, ak5);              // 5th stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + hstep * (
            b61 * dydx[i] + b62 * ak2[i] + 
            b63 * ak3[i] + b64 * ak4[i] + b65 * ak5[i]);
    }
    RightHandSide(yTemp, ak6);              // 6th stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yOut[i] = fyIn[i] + hstep * (
            b71 * dydx[i] + b72 * ak2[i] + b73 * ak3[i] +
            b74 * ak4[i] + b75 * ak5[i] + b76 * ak6[i]);
    }
    RightHandSide(yOut, ak7);               // 7th and Final stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yErr[i] = hstep * (
            dc1 * dydx[i] + dc2 * ak2[i] + 
            dc3 * ak3[i] + dc4 * ak4[i] +
            dc5 * ak5[i] + dc6 * ak6[i] + dc7 * ak7[i]
        ) + 1.5e-18;

        // Store Input and Final values, for possible use in calculating chord
        //
        fyOut[i] = yOut[i];
        fdydxIn[i] = dydx[i];        
    }
    
    fLastStepLength = hstep;
}

G4double G4DormandPrince745::DistChord() const
{
    // Coefficients were taken from Some Practical Runge-Kutta Formulas
    // by Lawrence F. Shampine, page 149, c*
    //
    const G4double hf1 = 6025192743.0 / 30085553152.0,
                   hf3 = 51252292925.0 / 65400821598.0,
                   hf4 = - 2691868925.0 / 45128329728.0,
                   hf5 = 187940372067.0 / 1594534317056.0,
                   hf6 = - 1776094331.0 / 19743644256.0,
                   hf7 = 11237099.0 / 235043384.0;

    G4ThreeVector mid;

    for(G4int i = 0; i < 3; ++i) 
    {
        mid[i] = fyIn[i] + 0.5 * fLastStepLength * (
            hf1 * fdydxIn[i] + hf3 * ak3[i] + 
            hf4 * ak4[i] + hf5 * ak5[i] + hf6 * ak6[i] + hf7 * ak7[i]);
    }

    const G4ThreeVector begin = makeVector(fyIn, Value3D::Position);
    const G4ThreeVector end = makeVector(fyOut, Value3D::Position);

    return G4LineSection::Distline(mid, begin, end);
}

// The lower (4th) order interpolant given by Dormand and Prince:
//        J. R. Dormand and P. J. Prince, "Runge-Kutta triples"
//        Computers & Mathematics with Applications, vol. 12, no. 9,
//        pp. 1007-1017, 1986.
//
void G4DormandPrince745::
Interpolate4thOrder(G4double yOut[], G4double tau) const
{
    const G4int numberOfVariables = GetNumberOfVariables();
    
    const G4double tau2 = tau * tau,
                   tau3 = tau * tau2,
                   tau4 = tau2 * tau2;

    const G4double bf1 = 1.0 / 11282082432.0 * (
        157015080.0 * tau4 - 13107642775.0 * tau3 + 34969693132.0 * tau2 - 
        32272833064.0 * tau + 11282082432.0);

    const G4double bf3 = - 100.0 / 32700410799.0 * tau * (
        15701508.0 * tau3 - 914128567.0 * tau2 + 2074956840.0 * tau - 
        1323431896.0);
    
    const G4double bf4 = 25.0 / 5641041216.0 * tau * (
        94209048.0 * tau3 - 1518414297.0 * tau2 + 2460397220.0 * tau - 
        889289856.0);
    
    const G4double bf5 = - 2187.0 / 199316789632.0 * tau * (
        52338360.0 * tau3 - 451824525.0 * tau2 + 687873124.0 * tau - 
        259006536.0);

    const G4double bf6 = 11.0 / 2467955532.0 * tau * (
        106151040.0 * tau3 - 661884105.0 * tau2 + 
        946554244.0 * tau - 361440756.0);
    
    const G4double bf7 = 1.0 / 29380423.0 * tau * (1.0 - tau) * (
        8293050.0 * tau2 - 82437520.0 * tau + 44764047.0);

    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yOut[i] = fyIn[i] + fLastStepLength * tau * (
            bf1 * fdydxIn[i] + bf3 * ak3[i] + bf4 * ak4[i] + 
            bf5 * ak5[i] + bf6 * ak6[i] + bf7 * ak7[i]);
    }
}

// Following interpolant of order 5 was given by Baker,Dormand,Gilmore, Prince :
//        T. S. Baker, J. R. Dormand, J. P. Gilmore, and P. J. Prince,
//        "Continuous approximation with embedded Runge-Kutta methods"
//        Applied Numerical Mathematics, vol. 22, no. 1, pp. 51-62, 1996.
//
// Calculating the extra stages for the interpolant
//
void G4DormandPrince745::SetupInterpolation5thOrder()
{
    // Coefficients for the additional stages
    //
    const G4double b81 = 6245.0 / 62208.0,
                   b82 = 0.0,
                   b83 = 8875.0 / 103032.0,
                   b84 = -125.0 / 1728.0,
                   b85 = 801.0 / 13568.0,
                   b86 = -13519.0 / 368064.0,
                   b87 = 11105.0 / 368064.0,
        
                   b91 = 632855.0 / 4478976.0,
                   b92 = 0.0,
                   b93 = 4146875.0 / 6491016.0,
                   b94 = 5490625.0 /14183424.0,
                   b95 = -15975.0 / 108544.0,
                   b96 = 8295925.0 / 220286304.0,
                   b97 = -1779595.0 / 62938944.0,
                   b98 = -805.0 / 4104.0;
    
    const G4int numberOfVariables = GetNumberOfVariables();
    State yTemp;

    // Evaluate the extra stages
    //
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + fLastStepLength * (
            b81 * fdydxIn[i] + b82 * ak2[i] + b83 * ak3[i] +
            b84 * ak4[i] + b85 * ak5[i] + b86 * ak6[i] +
            b87 * ak7[i]
        );
    }
    RightHandSide(yTemp, ak8);          // 8th Stage
    
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yTemp[i] = fyIn[i] + fLastStepLength * (
            b91 * fdydxIn[i] + b92 * ak2[i] + b93 * ak3[i] +
            b94 * ak4[i] + b95 * ak5[i] + b96 * ak6[i] +
            b97 * ak7[i] + b98 * ak8[i]
        );
    }
    RightHandSide(yTemp, ak9);          // 9th Stage
}

// Calculating the interpolated result yOut with the coefficients
//
void G4DormandPrince745::
Interpolate5thOrder(G4double yOut[], G4double tau) const
{
    // Define the coefficients for the polynomials
    //
    G4double bi[10][5];
    
    //  COEFFICIENTS OF   bi[1]
    bi[1][0] = 1.0,
    bi[1][1] = -38039.0 / 7040.0,
    bi[1][2] = 125923.0 / 10560.0,
    bi[1][3] = -19683.0 / 1760.0,
    bi[1][4] = 3303.0 / 880.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[2]
    bi[2][0] = 0.0,
    bi[2][1] = 0.0,
    bi[2][2] = 0.0,
    bi[2][3] = 0.0,
    bi[2][4] = 0.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[3]
    bi[3][0] = 0.0,
    bi[3][1] = -12500.0 / 4081.0,
    bi[3][2] = 205000.0 / 12243.0,
    bi[3][3] = -90000.0 / 4081.0,
    bi[3][4] = 36000.0 / 4081.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[4]
    bi[4][0] = 0.0,
    bi[4][1] = -3125.0 / 704.0,
    bi[4][2] = 25625.0 / 1056.0,
    bi[4][3] = -5625.0 / 176.0,
    bi[4][4] = 1125.0 / 88.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[5]
    bi[5][0] = 0.0,
    bi[5][1] = 164025.0 / 74624.0,
    bi[5][2] = -448335.0 / 37312.0,
    bi[5][3] = 295245.0 / 18656.0,
    bi[5][4] = -59049.0 / 9328.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[6]
    bi[6][0] = 0.0,
    bi[6][1] = -25.0 / 28.0,
    bi[6][2] = 205.0 / 42.0,
    bi[6][3] = -45.0 / 7.0,
    bi[6][4] = 18.0 / 7.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[7]
    bi[7][0] = 0.0,
    bi[7][1] = -2.0 / 11.0,
    bi[7][2] = 73.0 / 55.0,
    bi[7][3] = -171.0 / 55.0,
    bi[7][4] = 108.0 / 55.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[8]
    bi[8][0] = 0.0,
    bi[8][1] = 189.0 / 22.0,
    bi[8][2] = -1593.0 / 55.0,
    bi[8][3] = 3537.0 / 110.0,
    bi[8][4] = -648.0 / 55.0,
    //  --------------------------------------------------------
    //
    //  COEFFICIENTS OF  bi[9]
    bi[9][0] = 0.0,
    bi[9][1] = 351.0 / 110.0,
    bi[9][2] = -999.0 / 55.0,
    bi[9][3] = 2943.0 / 110.0,
    bi[9][4] = -648.0 / 55.0;
    //  --------------------------------------------------------
        
    // Calculating the polynomials
 
    G4double b[10];
    std::memset(b, 0.0, sizeof(b));

    G4double tauPower = 1.0;
    for(G4int j = 0; j <= 4; ++j)
    {       
       for(G4int iStage = 1; iStage <= 9; ++iStage)
       {
          b[iStage] += bi[iStage][j] * tauPower;
       }
       tauPower *= tau;       
    }
    
    const G4int numberOfVariables = GetNumberOfVariables();
    const G4double stepLen = fLastStepLength * tau;
    for(G4int i = 0; i < numberOfVariables; ++i)
    {
        yOut[i] = fyIn[i] + stepLen * (
            b[1] * fdydxIn[i] + b[2] * ak2[i] + b[3] * ak3[i] +
            b[4] * ak4[i] + b[5] * ak5[i] + b[6] * ak6[i] +
            b[7] * ak7[i] + b[8] * ak8[i] + b[9] * ak9[i]
        );
    }
}