rka.cpp

c++的数学物理方程数值算法源程序。这是"Numerical Methods for Physics"第二版的源程序。

#include "NumMeth.h"

void rk4(double x[], int nX, double t, double tau, 
  void (*derivsRK)(double x[], double t, double param[], double deriv[]), 
  double param[]);

void rka( double x[], int nX, double& t, double& tau, double err,
  void (*derivsRK)(double x[], double t, double param[], double deriv[]), 
  double param[]) {
// Adaptive Runge-Kutta routine
// Inputs
//   x          Current value of the dependent variable
//   nX         Number of elements in dependent variable x
//   t          Independent variable (usually time)
//   tau        Step size (usually time step)
//   err        Desired fractional local truncation error
//   derivsRK   Right hand side of the ODE; derivsRK is the
//              name of the function which returns dx/dt
//              Calling format derivsRK(x,t,param).
//   param      Extra parameters passed to derivsRK
// Outputs
//   x          New value of the dependent variable
//   t          New value of the independent variable
//   tau        Suggested step size for next call to rka

  //* Set initial variables
  double tSave = t;      // Save initial value
  double safe1 = 0.9, safe2 = 4.0;  // Safety factors

  //* Loop over maximum number of attempts to satisfy error bound
  double *xSmall, *xBig;
  xSmall = new double [nX+1];  xBig = new double [nX+1];
  int i, iTry, maxTry = 100;  
  for( iTry=1; iTry<=maxTry; iTry++ ) {
	
    //* Take the two small time steps
    double half_tau = 0.5 * tau;
    for( i=1; i<=nX; i++ )
      xSmall[i] = x[i];
    rk4(xSmall,nX,tSave,half_tau,derivsRK,param);
    t = tSave + half_tau;
    rk4(xSmall,nX,t,half_tau,derivsRK,param);
  
    //* Take the single big time step
    t = tSave + tau;
    for( i=1; i<=nX; i++ )
      xBig[i] = x[i];
    rk4(xBig,nX,tSave,tau,derivsRK,param);
  
    //* Compute the estimated truncation error
    double errorRatio = 0.0, eps = 1.0e-16;
    for( i=1; i<=nX; i++ ) {
      double scale = err * (fabs(xSmall[i]) + fabs(xBig[i]))/2.0;
      double xDiff = xSmall[i] - xBig[i];
      double ratio = fabs(xDiff)/(scale + eps);
      errorRatio = ( errorRatio > ratio ) ? errorRatio:ratio;
    }
    
    //* Estimate new tau value (including safety factors)
    double tau_old = tau;
    tau = safe1*tau_old*pow(errorRatio, -0.20);
    tau = (tau > tau_old/safe2) ? tau:tau_old/safe2;
    tau = (tau < safe2*tau_old) ? tau:safe2*tau_old;
  
    //* If error is acceptable, return computed values
    if (errorRatio < 1)  {
      for( i=1; i<=nX; i++ )
        x[i] = xSmall[i];
      return; 
    }
  }

  //* Issue error message if error bound never satisfied
  cout << "ERROR: Adaptive Runge-Kutta routine failed" << endl;
}