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/*
* abm_precision.cpp
*
* example to check the order of the multi-step methods
*
* Copyright 2009-2013 Karsten Ahnert
* Copyright 2009-2013 Mario Mulansky
*
* Distributed under the Boost Software License, Version 1.0.
* (See accompanying file LICENSE_1_0.txt or
* copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#include <iostream>
#include <cmath>
#include <boost/array.hpp>
#include <boost/numeric/odeint.hpp>
using namespace boost::numeric::odeint;
const int Steps = 4;
typedef double value_type;
typedef boost::array< double , 2 > state_type;
typedef runge_kutta_fehlberg78<state_type> initializing_stepper_type;
typedef adams_bashforth_moulton< Steps , state_type > stepper_type;
//typedef adams_bashforth< Steps , state_type > stepper_type;
// harmonic oscillator, analytic solution x[0] = sin( t )
struct osc
{
void operator()( const state_type &x , state_type &dxdt , const double t ) const
{
dxdt[0] = x[1];
dxdt[1] = -x[0];
}
};
int main()
{
stepper_type stepper;
initializing_stepper_type init_stepper;
const int o = stepper.order()+1; //order of the error is order of approximation + 1
const state_type x0 = {{ 0.0 , 1.0 }};
state_type x1 = x0;
double t = 0.0;
double dt = 0.25;
// initialization, does a number of steps already to fill internal buffer, t is increased
// we use the rk78 as initializing stepper
stepper.initialize( boost::ref(init_stepper) , osc() , x1 , t , dt );
// do a number of steps to fill the buffer with results from adams bashforth
for( size_t n=0 ; n < stepper.steps ; ++n )
{
stepper.do_step( osc() , x1 , t , dt );
t += dt;
}
double A = std::sqrt( x1[0]*x1[0] + x1[1]*x1[1] );
double phi = std::asin(x1[0]/A) - t;
// now we do the actual step
stepper.do_step( osc() , x1 , t , dt );
// only examine the error of the adams-bashforth-moulton step, not the initialization
const double f = 2.0 * std::abs( A*sin(t+dt+phi) - x1[0] ) / std::pow( dt , o ); // upper bound
std::cout << "# " << o << " , " << f << std::endl;
/* as long as we have errors above machine precision */
while( f*std::pow( dt , o ) > 1E-16 )
{
x1 = x0;
t = 0.0;
stepper.initialize( boost::ref(init_stepper) , osc() , x1 , t , dt );
A = std::sqrt( x1[0]*x1[0] + x1[1]*x1[1] );
phi = std::asin(x1[0]/A) - t;
// now we do the actual step
stepper.do_step( osc() , x1 , t , dt );
// only examine the error of the adams-bashforth-moulton step, not the initialization
std::cout << dt << '\t' << std::abs( A*sin(t+dt+phi) - x1[0] ) << std::endl;
dt *= 0.5;
}
}
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