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