G4BogackiShampine45.cc

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//
// G4BogackiShampine45 implementation
//
// Bogacki-Shampine's RK 5(4) non-FSAL interpolation method
// Definition of the stepper() method that evaluates one step in
// field propagation.
//
// The Butcher table of the Bogacki-Shampine-8-4-5 method is:
//
//   0  |
//  1/6 |      1/6
//  2/9 |      2/27          4/27
//  3/7 |    183/1372     -162/343      1053/1372
//  2/3 |     68/297        -4/11         42/143         1960/3861
//  3/4 |    597/22528      81/352     63099/585728     58653/366080    4617/20480
//   1  | 174197/959244 -30942/79937 8152137/19744439  666106/1039181 -29421/29068  482048/414219
//   1  |    587/8064         0      4440339/15491840   24353/124800     387/44800    2152/5985   7267/94080
//-------------------------------------------------------------------------------------------------------------------
//           587/8064         0      4440339/15491840   24353/124800     387/44800    2152/5985   7267/94080       0
//          2479/34992        0          123/416       612941/3411720    43/1440      2272/6561  79937/1113912  3293/556956
//
// Coefficients have been obtained from:
// http://www.netlib.org/ode/rksuite/
//
// Note on meaning of label "non-FSAL version": 
//   This method calculates the deriviative dy/dx at the endpoint of the
//   integration interval at each step, as part of its evaluation of the
//   endpoint and its error. So this value is available to be returned,
//   for re-use in case of a successful step.
//   (This is done in a 'later' version using a refined interface).
//
// Created: Somnath Banerjee, Google Summer of Code 2015, May-August 2015
// Revision: John Apostolakis, CERN, May 2016
// --------------------------------------------------------------------

#include <cassert>

#include "G4BogackiShampine45.hh"
#include "G4LineSection.hh"

G4bool G4BogackiShampine45::fPreparedConstants = false;
G4double G4BogackiShampine45::bi[12][7];

// Constructor
//
G4BogackiShampine45::G4BogackiShampine45(G4EquationOfMotion *EqRhs,
                                         G4int     noIntegrationVariables,
                                         G4bool    primary)   
   : G4MagIntegratorStepper(EqRhs, noIntegrationVariables)
{
    const G4int numberOfVariables = noIntegrationVariables;
    
    // New Chunk of memory being created for use by the stepper
    
    // aki - for storing intermediate RHS
    ak2 = new G4double[numberOfVariables];
    ak3 = new G4double[numberOfVariables];
    ak4 = new G4double[numberOfVariables];
    ak5 = new G4double[numberOfVariables];
    ak6 = new G4double[numberOfVariables];
    ak7 = new G4double[numberOfVariables];
    ak8 = new G4double[numberOfVariables];
    ak9 = new G4double[numberOfVariables];
    ak10 = new G4double[numberOfVariables];
    ak11 = new G4double[numberOfVariables];    

    for (auto i = 0; i < 6; ++i)
    {
       p[i]= new G4double[numberOfVariables];
    }

    assert ( GetNumberOfStateVariables() >= 8 );    
    const G4int numStateVars = std::max(noIntegrationVariables,
                                        GetNumberOfStateVariables() );  

    // Must ensure space extra 'state' variables exists - i.e. yIn[7]
    yTemp = new G4double[numStateVars];
    yIn = new G4double[numStateVars] ;
    
    fLastInitialVector = new G4double[numStateVars] ;
    fLastFinalVector = new G4double[numStateVars] ;
    fLastDyDx = new G4double[numberOfVariables];  // Only derivatives

    fMidVector = new G4double[numberOfVariables];
    fMidError =  new G4double[numberOfVariables];

    if( ! fPreparedConstants )
    {
       PrepareConstants();
    }
    
    if( primary )
    {
       fAuxStepper = new G4BogackiShampine45(EqRhs, numberOfVariables, false);
    }
}

// Destructor
//
G4BogackiShampine45::~G4BogackiShampine45()
{
    // Clear all previously allocated memory for stepper and DistChord
    //
    delete [] ak2;
    delete [] ak3;
    delete [] ak4;
    delete [] ak5;
    delete [] ak6;
    delete [] ak7;
    delete [] ak8;
    delete [] ak9;
    delete [] ak10;
    delete [] ak11;

    for (auto i = 0; i < 6; ++i)
    {
       delete [] p[i];
    }

    delete [] yTemp;
    delete [] yIn;
    
    delete [] fLastInitialVector;
    delete [] fLastFinalVector;
    delete [] fLastDyDx;
    delete [] fMidVector;
    delete [] fMidError;
    
    delete fAuxStepper;
}

void G4BogackiShampine45::GetLastDydx( G4double dyDxLast[] )
{
   const G4int numberOfVariables = GetNumberOfVariables();
   
   for(G4int i=0; i < numberOfVariables; ++i )
   {
      dyDxLast[i] = ak9[i];
   }
}       

// 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 G4BogackiShampine45::Stepper( const G4double yInput[],
                                   const G4double DyDx[],
                                         G4double Step,
                                         G4double yOut[],
                                         G4double yErr[] )
{
    G4int i;
    
    // Constants from the Butcher tableu
    //
    const G4double 
       b21 = 1.0/6.0 ,
       b31 = 2.0/27.0 ,     b32 = 4.0/27.0,
       
       b41 = 183.0/1372.0 , b42 = -162.0/343.0, b43 = 1053.0/1372.0,
       
       b51 = 68.0/297.0,    b52 = -4.0/11.0,
       b53 = 42.0/143.0,    b54 = 1960.0/3861.0,
       
       b61 = 597.0/22528.0,    b62 = 81.0/352.0,
       b63 = 63099.0/585728.0, b64 = 58653.0/366080.0,
       b65 = 4617.0/20480.0,
       
       b71 = 174197.0/959244.0,    b72 = -30942.0/79937.0,
       b73 = 8152137.0/19744439.0, b74 = 666106.0/1039181.0,
       b75 = -29421.0/29068.0,     b76 = 482048.0/414219.0,
       
       b81 = 587.0/8064.0,         b82 = 0.0,
       b83 = 4440339.0/15491840.0, b84 = 24353.0/124800.0,
       b85 = 387.0/44800.0,        b86 = 2152.0/5985.0,
       b87 = 7267.0/94080.0;
    
//    c1 = 2479.0/34992.0,
//    c2 = 0.0,
//    c3 = 123.0/416.0,
//    c4 = 612941.0/3411720.0,
//    c5 = 43.0/1440.0,
//    c6 = 2272.0/6561.0,
//    c7 = 79937.0/1113912.0,
//    c8 = 3293.0/556956.0,
    
    // For the embedded higher order method only the difference of values
    // taken and is used directly later (instead of defining the last row
    // of Butcher table in separate constants and taking the
    // difference)
    //
    const G4double 
       dc1 = b81 -   2479.0 /   34992.0 ,
       dc2 = 0.0,
       dc3 = b83 -    123.0 /     416.0 ,
       dc4 = b84 - 612941.0 / 3411720.0,
       dc5 = b85 -     43.0 /    1440.0,
       dc6 = b86 -   2272.0 /    6561.0,
       dc7 = b87 -  79937.0 / 1113912.0,
       dc8 =     -   3293.0 /  556956.0;

    const G4int numberOfVariables = GetNumberOfVariables();
    
    // The number of variables to be integrated over
    //
    yOut[7] = yTemp[7]  = yIn[7] = yInput[7];

    // Saving yInput because yInput and yOut can be aliases for same array
    //
    for(i=0; i<numberOfVariables; ++i)
    {
        yIn[i]=yInput[i];
    }
    
    // RightHandSide(yIn, dydx) ;
    // 1st Step - Not doing, getting passed
    //
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + b21*Step*DyDx[i] ;
    }
    RightHandSide(yTemp, ak2) ;              // 2nd Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(b31*DyDx[i] + b32*ak2[i]) ;
    }
    RightHandSide(yTemp, ak3) ;              // 3rd Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(b41*DyDx[i] + b42*ak2[i] + b43*ak3[i]) ;
    }
    RightHandSide(yTemp, ak4) ;              // 4th Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(b51*DyDx[i] + b52*ak2[i] + b53*ak3[i] +
                                  b54*ak4[i]) ;
    }
    RightHandSide(yTemp, ak5) ;              // 5th Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(b61*DyDx[i] + b62*ak2[i] + b63*ak3[i] +
                                  b64*ak4[i] + b65*ak5[i]) ;
    }
    RightHandSide(yTemp, ak6) ;              // 6th Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(b71*DyDx[i] + b72*ak2[i] + b73*ak3[i] +
                                  b74*ak4[i] + b75*ak5[i] + b76*ak6[i]);
    }
    RightHandSide(yTemp, ak7);               // 7th Step
    
    for(i=0; i<numberOfVariables; ++i)
    {
        yOut[i] = yIn[i] + Step*(b81*DyDx[i] + b82*ak2[i] + b83*ak3[i] +
                                  b84*ak4[i] + b85*ak5[i] + b86*ak6[i] +
                                  b87*ak7[i]);
    }
    RightHandSide(yOut, ak8);                // 8th Step - Final one Using FSAL

    for(i=0; i<numberOfVariables; ++i)
    {
        yErr[i] = Step*(dc1*DyDx[i] + dc2*ak2[i] + dc3*ak3[i] + dc4*ak4[i] +
                        dc5*ak5[i] + dc6*ak6[i] + dc7*ak7[i] + dc8*ak8[i]) ;
        
        // Store Input and Final values, for possible use in calculating chord
        //
        fLastInitialVector[i] = yIn[i] ;
        fLastFinalVector[i]   = yOut[i];
        fLastDyDx[i]          = DyDx[i];
    }
    
    fLastStepLength = Step;
    fPreparedInterpolation= false;

    return ;
}

// DistChord
//
G4double G4BogackiShampine45::DistChord() const
{
    G4double distLine, distChord;
    G4ThreeVector initialPoint, finalPoint, midPoint;
    
    // Store last initial and final points
    // (they will be overwritten in self-Stepper call!)
    //
    initialPoint = G4ThreeVector(fLastInitialVector[0],
                                 fLastInitialVector[1], fLastInitialVector[2]);
    finalPoint   = G4ThreeVector(fLastFinalVector[0],
                                 fLastFinalVector[1],  fLastFinalVector[2]);

#if 1
    // Old method -- Do half a step using StepNoErr
    //
    fAuxStepper->Stepper( fLastInitialVector, fLastDyDx, 0.5*fLastStepLength,
                          fMidVector, fMidError);
#else
    // New method -- Using interpolation,
    // requires only 3 extra stages (ie 3 extra field evaluations )

    // Use Interpolation, instead of auxiliary stepper to evaluate midpoint
    //
    if( ! fPreparedInterpolation )
    {
       G4BogackiShampine45* cThis = const_cast<G4BogackiShampine45 *>(this);
       cThis-> SetupInterpolationHigh();
    }
    // For calculating the output at the tau fraction of Step
    //
    G4double tau = 0.5;
    InterpolateHigh( tau, fMidVector );    
#endif
   
    midPoint = G4ThreeVector( fMidVector[0], fMidVector[1], fMidVector[2]);
    
    // Use stored values of Initial and Endpoint + new Midpoint to evaluate
    // distance of Chord
    
    if (initialPoint != finalPoint)
    {
        distLine  = G4LineSection::Distline( midPoint,initialPoint,finalPoint );
        distChord = distLine;
    }
    else
    {
        distChord = (midPoint-initialPoint).mag();
    }
    return distChord;
}

void G4BogackiShampine45::SetupInterpolationHigh()
{
    // Coefficients for the additional stages
    //
    const G4double
    a91 = 455.0/6144.0 ,
    a92 = 0.0 ,
    a93 = 10256301.0/35409920.0 ,
    a94 = 2307361.0/17971200.0 ,
    a95 = -387.0/102400.0 ,
    a96 = 73.0/5130.0 ,
    a97 = -7267.0/215040.0 ,
    a98 = 1.0/32.0 ,
    
    a101 = -837888343715.0/13176988637184.0 ,
    a102 = 30409415.0/52955362.0 ,
    a103 = -48321525963.0/759168069632.0 ,
    a104 = 8530738453321.0/197654829557760.0 ,
    a105 = 1361640523001.0/1626788720640.0 ,
    a106 = -13143060689.0/38604458898.0 ,
    a107 = 18700221969.0/379584034816.0 ,
    a108 = -5831595.0/847285792.0 ,
    a109 = -5183640.0/26477681.0 ,
    
    a111 = 98719073263.0/1551965184000.0 ,
    a112 = 1307.0/123552.0 ,
    a113 = 4632066559387.0/70181753241600.0 ,
    a114 = 7828594302389.0/382182512025600.0 ,
    a115 = 40763687.0/11070259200.0 ,
    a116 = 34872732407.0/224610586200.0 ,
    a117 = -2561897.0/30105600.0 ,
    a118 = 1.0/10.0 ,
    a119 = -1.0/10.0 ,
    a1110 = -1403317093.0/11371610250.0 ;
    
    const G4int numberOfVariables= this->GetNumberOfVariables();
    const G4double* dydx= fLastDyDx;
    const G4double Step = fLastStepLength;
    
    yTemp[7]  = yIn[7];

    // Evaluate the extra stages
    //
    for(G4int i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(a91*dydx[i] + a92*ak2[i] + a93*ak3[i] +
                                  a94*ak4[i] + a95*ak5[i] + a96*ak6[i] +
                                  a97*ak7[i] + a98*ak8[i] );
    }
    
    RightHandSide(yTemp, ak9);   // 9th stage
    
    for(G4int i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(a101*dydx[i] + a102*ak2[i] + a103*ak3[i] +
                                  a104*ak4[i] + a105*ak5[i] + a106*ak6[i] +
                                  a107*ak7[i] + a108*ak8[i] + a109*ak9[i] );
    }
    
    RightHandSide(yTemp, ak10);  // 10th stage
    
    for(G4int i=0; i<numberOfVariables; ++i)
    {
        yTemp[i] = yIn[i] + Step*(a111*dydx[i] + a112*ak2[i] + a113*ak3[i] +
                                  a114*ak4[i] + a115*ak5[i] + a116*ak6[i] +
                                  a117*ak7[i] + a118*ak8[i] + a119*ak9[i] +
                                  a1110*ak10[i] );
    }
    RightHandSide(yTemp, ak11);  // 11th stage

    //  In future we can restrict the number of variables interpolated
    //
    G4int nwant = numberOfVariables; 

    //  Form the coefficients of the interpolating polynomial in its shifted
    //  and scaled form.  The terms are grouped to minimize the errors 
    //  of the transformation, to cope with ill-conditioning. ( From RKSUITE )
    //   
    for (G4int l = 0; l < nwant; ++l)
    {
        //  Coefficient of tau^6        
        p[5][l] =   bi[5][6]*ak5[l] +
                  ((bi[10][6]*ak10[l] + bi[8][6]*ak8[l]) + 
                   (bi[7][6]*ak7[l] + bi[6][6]*ak6[l]))  + 
                  ((bi[4][6]*ak4[l] + bi[9][6]*ak9[l]) + 
                   (bi[3][6]*ak3[l] + bi[11][6]*ak11[l]) + 
                    bi[1][6]*dydx[l]);
        //  Coefficient of tau^5
        p[4][l] = (bi[10][5]*ak10[l] + bi[9][5]*ak9[l])  + 
                 ((bi[7][5]*ak7[l] + bi[6][5]*ak6[l]) + 
                   bi[5][5]*ak5[l])  +  ((bi[4][5]*ak4[l] + 
                   bi[8][5]*ak8[l]) + (bi[3][5]*ak3[l] + 
                   bi[11][5]*ak11[l]) + bi[1][5]*dydx[l]);
        //  Coefficient of tau^4
        p[3][l] = ((bi[4][4]*ak4[l] + bi[8][4]*ak8[l]) + 
                   (bi[7][4]*ak7[l] + bi[6][4]*ak6[l]) + 
                    bi[5][4]*ak5[l]) + ((bi[10][4]*ak10[l] + 
                    bi[9][4]*ak9[l]) +  (bi[3][4]*ak3[l] + 
                    bi[11][4]*ak11[l]) + bi[1][4]*dydx[l]);
        //  Coefficient of tau^3
        p[2][l] =  bi[5][3]*ak5[l] + bi[6][3]*ak6[l]  + 
                 ((bi[3][3]*ak3[l] + bi[9][3]*ak9[l]) + 
                 (bi[10][3]*ak10[l]+ bi[8][3]*ak8[l]) + bi[1][3]*dydx[l]) + 
                 ((bi[4][3]*ak4[l] + bi[11][3]*ak11[l]) + bi[7][3]*ak7[l]);
        //  Coefficient of tau^2
        p[1][l] = bi[5][2]*ak5[l]  + ((bi[6][2]*ak6[l] + 
                  bi[8][2]*ak8[l]) +   bi[1][2]*dydx[l])  + 
                ((bi[3][2]*ak3[l]  +   bi[9][2]*ak9[l]) + 
                 bi[10][2]*ak10[l])+ ((bi[4][2]*ak4[l] + 
                 bi[11][2]*ak2[l]) +   bi[7][2]*ak7[l]);
      }

      //  Scale all the coefficients by the step size.
      //
      for (G4int i = 0; i < 6; ++i)
      {
         for (G4int l = 0; l < nwant; ++l)
         {  
            p[i][l] *= Step;
         }
      }

    fPreparedInterpolation = true;
}

void G4BogackiShampine45::PrepareConstants()
{
    for(auto i=1; i<= 11; ++i)
    {
        bi[i][1] = 0.0 ;
    }
    
    for(auto i=1; i<=6; ++i)
    {
        bi[2][i] = 0.0 ;
    }

    bi[1][6] = -12134338393.0 / 1050809760.0 ,
    bi[1][5] =  -1620741229.0 / 50038560.0 ,
    bi[1][4] =  -2048058893.0 / 59875200.0 ,
    bi[1][3] = -87098480009.0 / 5254048800.0 ,
    bi[1][2] = -11513270273.0 / 3502699200.0 ,
    // 
    bi[3][6] =  -33197340367.0 / 1218433216.0 ,
    bi[3][5] = -539868024987.0 / 6092166080.0 ,
    bi[3][4] =  -39991188681.0 / 374902528.0 ,
    bi[3][3] =  -69509738227.0 / 1218433216.0 ,
    bi[3][2] =  -29327744613.0 / 2436866432.0 ,
    //
    bi[4][6] =   -284800997201.0 /  19905339168.0 ,
    bi[4][5] =  -7896875450471.0 / 165877826400.0 ,
    bi[4][4] =   -333945812879.0 /   5671036800.0 ,
    bi[4][3] = -16209923456237.0 / 497633479200.0 ,
    bi[4][2] =  -2382590741699.0 / 331755652800.0 ,
    //
    bi[5][6] = -540919.0 / 741312.0 ,
    bi[5][5] = -103626067.0 / 43243200.0 ,
    bi[5][4] = -633779.0 / 211200.0 ,
    bi[5][3] = -32406787.0 / 18532800.0 ,
    bi[5][2] = -36591193.0 / 86486400.0 ,
    //
    bi[6][6] = 7157998304.0 / 374350977.0 ,
    bi[6][5] = 30405842464.0 / 623918295.0 ,
    bi[6][4] = 183022264.0 / 5332635.0 ,
    bi[6][3] = -3357024032.0 / 1871754885.0 ,
    bi[6][2] = -611586736.0 / 89131185.0 ,
    //
    bi[7][6] =  -138073.0 /  9408.0 ,
    bi[7][5] =  -719433.0 / 15680.0 ,
    bi[7][4] = -1620541.0 / 31360.0 ,
    bi[7][3] =  -385151.0 / 15680.0 ,
    bi[7][2] =  -65403.0  / 15680.0 ,
    //
    bi[8][6] = 1245.0 / 64.0 ,
    bi[8][5] = 3991.0 / 64.0 ,
    bi[8][4] = 4715.0 / 64.0 ,
    bi[8][3] = 2501.0 / 64.0 ,
    bi[8][2] =  149.0 / 16.0 ,
    bi[8][1] = 1.0 ,
    //
    bi[9][6] = 55.0 / 3.0 ,
    bi[9][5] = 71.0 ,
    bi[9][4] = 103.0 ,
    bi[9][3] = 199.0 / 3.0 ,
    bi[9][2] = 16.0 ,
    //
    bi[10][6] =  -1774004627.0  /  75810735.0 ,
    bi[10][5] =  -1774004627.0  /  25270245.0 ,
    bi[10][4] =    -26477681.0  /    359975.0 ,
    bi[10][3] = -11411880511.0  / 379053675.0 ,
    bi[10][2] =   -423642896.0  / 126351225.0 ,
    //
    bi[11][6] =  35.0 ,
    bi[11][5] = 105.0 ,
    bi[11][4] = 117.0 ,
    bi[11][3] =  59.0 ,
    bi[11][2] =  12.0 ;

    fPreparedConstants = true;
}

void G4BogackiShampine45::InterpolateHigh(G4double tau, G4double* yOut) const
{
    G4int numberOfVariables = GetNumberOfVariables();
    
    G4Exception("G4BogackiShampine45::InterpolateHigh()", "GeomField0001",
                FatalException, "Method is not yet validated.");

    // const G4double *yIn=  fLastInitialVector;
    // const G4double *dydx= fLastDyDx;
    const G4double Step = fLastStepLength;
    
#if 1
    G4int nwant = numberOfVariables;
    const G4int norder= 6;
    G4int l, k;

    for (l = 0; l < nwant; ++l)
    {
      yOut[l] = p[norder-1][l] * tau;
    }
    for (k = norder - 2; k >= 1; --k)
    {
      for (l = 0; l < nwant; ++l)
      {
         yOut[l] = ( yOut[l] + p[k][l] ) * tau;
      }
    }
    for (l = 0; l < nwant; ++l)
    {
      yOut[l] = ( yOut[l] + Step * ak8[l] ) * tau + yIn[l];
    }
    // The derivative at the end-point is nextDydx[i] = ak8[i];
#else
    // The scheme tries to do the same as the DormandPrince745 routine,
    // but fails 

    G4double b[12];
    const G4double* dydx = fLastDyDx;
    
    G4double tau0 = tau;
    
    for(G4int iStage=1; iStage<=11; ++iStage)   // iStage = stage number
    {
        b[iStage] = 0.0;
        tau = tau0;
        for(G4int j=6; j>=1; --j)               // j reversed
        {
            b[iStage] += bi[iStage][j] * tau;
            tau *= tau0;
        }
    }

    for(G4int i=0; i<numberOfVariables; ++i)
    {
        yOut[i] = yIn[i] + Step*(b[1]*dydx[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] +
                                 b[10]*ak10[i] + b[11]*ak11[i] );
    }
#endif    
}