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Lattice systems

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 = Σ​i p​i2/2 + 1/2 ( q​i+1 - q​i )^2 + β / 4 ( q​i+1 - q​i )^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 dq​i / dt = p​i. 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


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