G4TDormandPrince45.hh

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// G4TDormandPrince45
//
// Class desription:
//
//  An implementation of the 5th order embedded RK method from the paper:
//  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.
//
//  DormandPrince7 - 5(4) embedded RK method
 
// Author: Josh Xie (CERN, Google Summer of Code 2014), June 2014
// Supervisors:  Sandro Wenzel, John Apostolakis (CERN)
// --------------------------------------------------------------------
#ifndef G4TDORMAND_PRINCE_45_HH
#define G4TDORMAND_PRINCE_45_HH
 
#include "G4MagIntegratorStepper.hh"
#include "G4FieldUtils.hh"
#include "G4LineSection.hh"
 
#include <cstring>
#include <cassert>
 
/**
 * @brief G4TDormandPrince45 is a templated version of G4DormandPrince745
 * 5th order Runge-Kutta stepper.
 */
 
template <class T_Equation, unsigned int N = 6 >
class G4TDormandPrince45 : public G4MagIntegratorStepper
{
  public:
 
    G4TDormandPrince45(T_Equation* equation );
    G4TDormandPrince45(T_Equation* equation, G4int numVar ); // must have numVar == N
   
    inline void StepWithError(const G4double yInput[],
                              const G4double dydx[],
                                    G4double hstep,
                                    G4double yOutput[],
                                    G4double yError[] ) ;
   
    void Stepper(const G4double yInput[],
                 const G4double dydx[],
                       G4double hstep,
                       G4double yOutput[],
                       G4double yError[]) final;
 
    inline void StepWithFinalDerivate(const G4double yInput[],   
                                      const G4double dydx[],
                                            G4double hstep,
                                            G4double yOutput[],
                                            G4double yError[],
                                            G4double dydxOutput[]);
   
    inline void SetupInterpolation() {}
 
    void Interpolate(G4double tau, G4double yOut[]) const
    {
      Interpolate4thOrder(yOut, tau);
    }
      // For calculating the output at the tau fraction of Step
 
    G4double DistChord() const final;
 
    inline G4int IntegratorOrder() const override { return 4; }
 
    G4StepperType StepperType() const override { return kTDormandPrince45; }
 
    inline const field_utils::ShortState<N>& GetYOut() const { return fyOut; }
 
    void Interpolate4thOrder(G4double yOut[], G4double tau) const;
 
    void SetupInterpolation5thOrder();
    void Interpolate5thOrder(G4double yOut[], G4double tau) const;
  
    // __attribute__((always_inline))
    inline void RightHandSideInl( const G4double y[],
                                        G4double dydx[] )
    {
      fEquation_Rhs->T_Equation::RightHandSide(y, dydx);
    }
 
    inline void Stepper(const G4double yInput[],
                        const G4double dydx[],
                              G4double hstep, G4double yOutput[],
                              G4double yError[], G4double dydxOutput[])
    {
      StepWithFinalDerivate(yInput, dydx, hstep, yOutput, yError, dydxOutput);
    }
 
    T_Equation* GetSpecificEquation() { return fEquation_Rhs; }
   
    static constexpr G4int N8 = N > 8 ? N : 8;  //  y[
   
  private:
 
    field_utils::ShortState<N>  ak2, ak3, ak4, ak5, ak6, ak7, ak8, ak9; 
    field_utils::ShortState<N8> fyIn;
    field_utils::ShortState<N>  fyOut, fdydxIn;
 
    // - Simpler :
    // field_utils::State ak2, ak3, ak4, ak5, ak6, ak7, ak8, ak9;
    // field_utils::State fyIn, fyOut, fdydxIn;
 
    G4double fLastStepLength = -1.0;
    T_Equation* fEquation_Rhs;
};
 
// --------------------------------------------------------------------
// G4TDormandPrince745 implementation -- borrowed from G4DormandPrince745
//
// 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
//
// --------------------------------------------------------------------
 
// Constructor
//
template <class T_Equation, unsigned int N>
G4TDormandPrince45<T_Equation,N>::G4TDormandPrince45(T_Equation* equation )
    : G4MagIntegratorStepper(dynamic_cast<G4EquationOfMotion*>(equation), N )
    , fEquation_Rhs(equation)
{
  // assert( dynamic_cast<G4EquationOfMotion*>(equation) != nullptr );
  if( dynamic_cast<G4EquationOfMotion*>(equation) == nullptr )
  {
    G4Exception("G4TDormandPrince745: constructor", "GeomField0001",
                FatalException, "T_Equation is not an G4EquationOfMotion.");
  }
 
  /***
  assert( equation->GetNumberOfVariables == N );
  if( equation->GetNumberOfVariables != N ){
    G4ExceptionDescription msg;
    msg << "Equation has an incompatible number of variables." ;
    msg << "   template N = " << N << " equation-Nvar= "
        << equation->GetNumberOfVariables;
    G4Exception("G4TCashKarpRKF45: constructor", "GeomField0001",
                FatalException, msg );
   } ****/
}
 
template <class T_Equation, unsigned int N>
inline G4TDormandPrince45<T_Equation,N>::
       G4TDormandPrince45(T_Equation* equation, G4int numVar )
  : G4TDormandPrince45<T_Equation,N>(equation )
{
  if( numVar != G4int(N))
  {
    G4ExceptionDescription msg;
    msg << "Equation has an incompatible number of variables." ;
    msg << "   template N = " << N
        << "   argument numVar = " << numVar ;
    //    << " equation-Nvar= " << equation->GetNumberOfVariables(); // --> Expected later
    G4Exception("G4TCashKarpRKF45: constructor", "GeomField0001",
                   FatalErrorInArgument, msg );
  }
  assert( numVar == N );
}
 
template <class T_Equation, unsigned int N>
inline void
G4TDormandPrince45<T_Equation,N>::StepWithFinalDerivate(const G4double yInput[],
                                                        const G4double dydx[],
                                                              G4double hstep,
                                                              G4double yOutput[],
                                                              G4double yError[],
                                                              G4double dydxOutput[])
{
  StepWithError(yInput, dydx, hstep, yOutput, yError);
  field_utils::copy(dydxOutput, ak7, N);
}
 
// 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
//
 
template <class T_Equation, unsigned int N>
inline void
G4TDormandPrince45<T_Equation,N>::StepWithError(const G4double yInput[],
                                                const G4double dydx[],
                                                      G4double hstep,
                                                      G4double yOut[],
                                                      G4double yErr[] )
{
  // The parameters of the Butcher tableu
  //
  constexpr 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();
  //   The number of variables to be integrated over    
  field_utils::ShortState<N8> yTemp;
    
  yOut[7] = yTemp[7]  = fyIn[7] = yInput[7];  // Pass along the time - used in RightHandSide
    
  //  Saving yInput because yInput and yOut can be aliases for same array
  //
  for(unsigned int i = 0; i < N; ++i)
  {
    fyIn[i]  = yInput[i];
    yTemp[i] = yInput[i] + b21 * hstep * dydx[i];
  }
  RightHandSideInl(yTemp, ak2);              // 2nd stage
 
  for(unsigned int i = 0; i < N; ++i)
  {
    yTemp[i] = fyIn[i] + hstep * (b31 * dydx[i] + b32 * ak2[i]);
  }
  RightHandSideInl(yTemp, ak3);              // 3rd stage
    
  for(unsigned int i = 0; i < N; ++i)
  {
    yTemp[i] = fyIn[i] + hstep * (
        b41 * dydx[i] + b42 * ak2[i] + b43 * ak3[i]);
  }
  RightHandSideInl(yTemp, ak4);              // 4th stage
    
  for(unsigned int i = 0; i < N; ++i)
  {
    yTemp[i] = fyIn[i] + hstep * (
        b51 * dydx[i] + b52 * ak2[i] + b53 * ak3[i] + b54 * ak4[i]);
  }
  RightHandSideInl(yTemp, ak5);              // 5th stage
    
  for(unsigned int i = 0; i < N; ++i)
  {
    yTemp[i] = fyIn[i] + hstep * (
        b61 * dydx[i] + b62 * ak2[i] + 
        b63 * ak3[i] + b64 * ak4[i] + b65 * ak5[i]);
  }
  RightHandSideInl(yTemp, ak6);              // 6th stage
    
  for(unsigned int i = 0; i < N; ++i)
  {
    yOut[i] = fyIn[i] + hstep * (
        b71 * dydx[i] + b72 * ak2[i] + b73 * ak3[i] +
        b74 * ak4[i] + b75 * ak5[i] + b76 * ak6[i]);
  }
  RightHandSideInl(yOut, ak7);               // 7th and Final stage
    
  for(unsigned int i = 0; i < N; ++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;
}
 
template <class T_Equation, unsigned int N >
inline void
G4TDormandPrince45<T_Equation,N>::Stepper(const G4double yInput[],
                                          const G4double dydx[],
                                                G4double Step,
                                                G4double yOutput[],
                                                G4double yError[])
{
  assert( yOutput != yInput );
  assert( yError  != yInput );
  
  StepWithError( yInput, dydx, Step, yOutput, yError);
}
 
template <class T_Equation, unsigned int N>
inline G4double G4TDormandPrince45<T_Equation,N>::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(unsigned int 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, field_utils::Value3D::Position);
  const G4ThreeVector end = makeVector(fyOut,  field_utils::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.
//
template <class T_Equation, unsigned int N>
inline void
G4TDormandPrince45<T_Equation,N>::Interpolate4thOrder(G4double yOut[],
                                                      G4double tau) const
{    
  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(unsigned int i = 0; i < N; ++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
//
template <class T_Equation, unsigned int N>
inline void G4TDormandPrince45<T_Equation,N>::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;
 
  field_utils::ShortState<N> yTemp;
 
  // Evaluate the extra stages
  //
  for(unsigned int i = 0; i < N; ++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]
    );
  }
  RightHandSideInl(yTemp, ak8);          // 8th Stage
    
  for(unsigned int i = 0; i < N; ++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]
    );
  }
  RightHandSideInl(yTemp, ak9);          // 9th Stage
}
 
// Calculating the interpolated result yOut with the coefficients
//
template <class T_Equation, unsigned int N>
inline void G4TDormandPrince45<T_Equation,N>::
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 G4double stepLen = fLastStepLength * tau;
  for(G4int i = 0; i < N; ++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]
    );
  }
}
 
#endif