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odeint can also be used to solve ordinary differential equations defined on lattices. A prominent example is the Fermi-Pasta-Ulam system [8] . It is a Hamiltonian system of nonlinear coupled harmonic oscillators. The Hamiltonian is
H = Σi pi2/2 + 1/2 ( qi+1 - qi )^2 + β / 4 ( qi+1 - qi )^4
Remarkably, the Fermi-Pasta-Ulam system was the first numerical experiment to be implemented on a computer. It was studied at Los Alamos in 1953 on one of the first computers (a MANIAC I) and it triggered a whole new tree of mathematical and physical science.
Like the Solar System, the FPU is solved again by a symplectic solver, but in this case we can speed up the computation because the q components trivially reduce to dqi / dt = pi. odeint is capable of doing this performance improvement. All you have to do is to call the symplectic solver with an state function for the p components. Here is how this function looks like
typedef vector< double > container_type; struct fpu { const double m_beta; fpu( const double beta = 1.0 ) : m_beta( beta ) { } // system function defining the ODE void operator()( const container_type &q , container_type &dpdt ) const { size_t n = q.size(); double tmp = q[0] - 0.0; double tmp2 = tmp + m_beta * tmp * tmp * tmp; dpdt[0] = -tmp2; for( size_t i=0 ; i<n-1 ; ++i ) { tmp = q[i+1] - q[i]; tmp2 = tmp + m_beta * tmp * tmp * tmp; dpdt[i] += tmp2; dpdt[i+1] = -tmp2; } tmp = - q[n-1]; tmp2 = tmp + m_beta * tmp * tmp * tmp; dpdt[n-1] += tmp2; } // calculates the energy of the system double energy( const container_type &q , const container_type &p ) const { // ... } // calculates the local energy of the system void local_energy( const container_type &q , const container_type &p , container_type &e ) const { // ... } };
You can also use boost::array< double , N >
for the state type.
Now, you have to define your initial values and perform the integration:
const size_t n = 64; container_type q( n , 0.0 ) , p( n , 0.0 ); for( size_t i=0 ; i<n ; ++i ) { p[i] = 0.0; q[i] = 32.0 * sin( double( i + 1 ) / double( n + 1 ) * M_PI ); } const double dt = 0.1; typedef symplectic_rkn_sb3a_mclachlan< container_type > stepper_type; fpu fpu_instance( 8.0 ); integrate_const( stepper_type() , fpu_instance , make_pair( boost::ref( q ) , boost::ref( p ) ) , 0.0 , 1000.0 , dt , streaming_observer( cout , fpu_instance , 10 ) );
The observer uses a reference to the system object to calculate the local energies:
struct streaming_observer { std::ostream& m_out; const fpu &m_fpu; size_t m_write_every; size_t m_count; streaming_observer( std::ostream &out , const fpu &f , size_t write_every = 100 ) : m_out( out ) , m_fpu( f ) , m_write_every( write_every ) , m_count( 0 ) { } template< class State > void operator()( const State &x , double t ) { if( ( m_count % m_write_every ) == 0 ) { container_type &q = x.first; container_type &p = x.second; container_type energy( q.size() ); m_fpu.local_energy( q , p , energy ); for( size_t i=0 ; i<q.size() ; ++i ) { m_out << t << "\t" << i << "\t" << q[i] << "\t" << p[i] << "\t" << energy[i] << "\n"; } m_out << "\n"; clog << t << "\t" << accumulate( energy.begin() , energy.end() , 0.0 ) << "\n"; } ++m_count; } };
The full cpp file for this FPU example can be found here fpu.cpp