void do_step( System system , state_type& x , time_type t , time_type dt )
{
typedef typename odeint::unwrap_reference< System >::type system_type;
typedef typename odeint::unwrap_reference< typename system_type::first_type >::type deriv_func_type;
typedef typename odeint::unwrap_reference< typename system_type::second_type >::type jacobi_func_type;
system_type &sys = system;
deriv_func_type &deriv_func = sys.first;
jacobi_func_type &jacobi_func = sys.second;
m_resizer.adjust_size( x , detail::bind( &stepper_type::template resize_impl<state_type> , detail::ref( *this ) , detail::_1 ) );
state_type x1 = x;// x1 for storing x(k+1)
// std::cout << x1[0] << " " << x1[1] << "\n";
implicit_euler< double > imp_euler;
imp_euler.do_step(system, x1, t, dt);
//x(k+1) computed from above line for x(k) = 4*x(k+1) - 3*x(k+2) + 2*dt*x'(k+2)
// std::cout << x1[0] << " " << x1[1] << "\n";
for( size_t i=0 ; i<x.size() ; ++i )
m_pm.m_v[i] = i;
t +=dt;// t = t0 + 2*dt for x(k+2) and x'(k+2)
deriv_func( x1 , m_dxdt.m_v , t ); //m_dxdt.m_v is x'(k+2)
m_b.m_v = 2.0 * dt * m_dxdt.m_v;
jacobi_func( x1 , m_jacobi.m_v , t );
m_jacobi.m_v *= dt;
m_jacobi.m_v -= boost::numeric::ublas::identity_matrix< value_type >( x.size() );
solve( m_b.m_v , m_jacobi.m_v );
m_x.m_v = (4.0 *x1)- (3.0*x) + m_b.m_v;//x(k) after first Newton iteration
std::cout << "Before loop : " << m_x.m_v[0] << " " << m_x.m_v[1] << " " << m_b.m_v[0] << " " << m_b.m_v[1] << "\n";
while( boost::numeric::ublas::norm_2( m_b.m_v ) > m_epsilon )
{
deriv_func( m_x.m_v , m_dxdt.m_v , t );
m_b.m_v = -(4.0 *x1)+ (3.0*x) + m_x.m_v + 2.0 * dt * m_dxdt.m_v;
solve( m_b.m_v , m_jacobi.m_v );
m_x.m_v -= m_b.m_v;
std::cout << "Inside loop : " << m_x.m_v[0] << " " << m_x.m_v[1] << " " << m_b.m_v[0] << " " << m_b.m_v[1] << "\n";
}
x = m_x.m_v;
}
void do_step( System system , state_type &x , time_type t , time_type dt )
{
typedef typename odeint::unwrap_reference< System >::type system_type;
typedef typename odeint::unwrap_reference< typename system_type::first_type >::type deriv_func_type;
typedef typename odeint::unwrap_reference< typename system_type::second_type >::type jacobi_func_type;
system_type &sys = system;
deriv_func_type &deriv_func = sys.first;
jacobi_func_type &jacobi_func = sys.second;
m_resizer.adjust_size( x , detail::bind( &stepper_type::template resize_impl<state_type> , detail::ref( *this ) , detail::_1 ) );
for( size_t i=0 ; i<x.size() ; ++i )
m_pm.m_v[i] = i;
t += dt;
// apply first Newton step
deriv_func( x , m_dxdt.m_v , t );
m_b.m_v = dt * m_dxdt.m_v;
jacobi_func( x , m_jacobi.m_v , t );
m_jacobi.m_v *= dt;
m_jacobi.m_v -= boost::numeric::ublas::identity_matrix< value_type >( x.size() );
solve( m_b.m_v , m_jacobi.m_v );
m_x.m_v = x - m_b.m_v;
// iterate Newton until some precision is reached
// ToDo: maybe we should apply only one Newton step -> linear implicit one-step scheme
while( boost::numeric::ublas::norm_2( m_b.m_v ) > m_epsilon )
{
deriv_func( m_x.m_v , m_dxdt.m_v , t );
m_b.m_v = x - m_x.m_v + dt*m_dxdt.m_v;
// simplified version, only the first Jacobian is used
// jacobi( m_x , m_jacobi , t );
// m_jacobi *= dt;
// m_jacobi -= boost::numeric::ublas::identity_matrix< value_type >( x.size() );
solve( m_b.m_v , m_jacobi.m_v );
m_x.m_v -= m_b.m_v;
}
x = m_x.m_v;
}
void do_step( System system , state_type &x , time_type t , time_type dt )
{
typedef typename omplext_odeint::unwrap_reference< System >::type system_type;
typedef typename omplext_odeint::unwrap_reference< typename system_type::first_type >::type deriv_func_type;
typedef typename omplext_odeint::unwrap_reference< typename system_type::second_type >::type jacobi_func_type;
system_type &sys = system;
deriv_func_type &deriv_func = sys.first;
jacobi_func_type &jacobi_func = sys.second;
m_resizer.adjust_size( x , detail::bind(
&stepper_type::template resize_impl< state_type > , detail::ref( *this ) , detail::_1 ) );
m_identity.m_v = 1;
t += dt;
m_x.m_v = x;
deriv_func( x , m_dxdt.m_v , t );
jacobi_func( x , m_jacobi.m_v , t );
m_dxdt.m_v *= -dt;
m_jacobi.m_v *= dt;
m_jacobi.m_v -= m_identity.m_v ;
// using ilu_0 preconditioning -incomplete LU factorisation
// itl::pc::diagonal<matrix_type,double> L(m_jacobi.m_v);
itl::pc::ilu_0<matrix_type> L( m_jacobi.m_v );
solve( m_jacobi.m_v , m_x.m_v , m_dxdt.m_v , L );
x+= m_x.m_v;
}
void do_step( System system , const state_type &x , time_type t , state_type &xout , time_type dt , state_type &xerr )
{
// get the systen and jacobi function
typedef typename odeint::unwrap_reference< System >::type system_type;
typedef typename odeint::unwrap_reference< typename system_type::first_type >::type deriv_func_type;
typedef typename odeint::unwrap_reference< typename system_type::second_type >::type jacobi_func_type;
system_type &sys = system;
deriv_func_type &deriv_func = sys.first;
jacobi_func_type &jacobi_func = sys.second;
const size_t n = x.size();
m_resizer.adjust_size( x , detail::bind( &stepper_type::template resize_impl<state_type> , detail::ref( *this ) , detail::_1 ) );
for( size_t i=0 ; i<n ; ++i )
m_pm.m_v( i ) = i;
deriv_func( x , m_dxdt.m_v , t );
jacobi_func( x , m_jac.m_v , t , m_dfdt.m_v );
m_jac.m_v *= -1.0;
m_jac.m_v += 1.0 / m_coef.gamma / dt * boost::numeric::ublas::identity_matrix< value_type >( n );
boost::numeric::ublas::lu_factorize( m_jac.m_v , m_pm.m_v );
for( size_t i=0 ; i<n ; ++i )
m_g1.m_v[i] = m_dxdt.m_v[i] + dt * m_coef.d1 * m_dfdt.m_v[i];
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , m_g1.m_v );
for( size_t i=0 ; i<n ; ++i )
m_xtmp.m_v[i] = x[i] + m_coef.a21 * m_g1.m_v[i];
deriv_func( m_xtmp.m_v , m_dxdtnew.m_v , t + m_coef.c2 * dt );
for( size_t i=0 ; i<n ; ++i )
m_g2.m_v[i] = m_dxdtnew.m_v[i] + dt * m_coef.d2 * m_dfdt.m_v[i] + m_coef.c21 * m_g1.m_v[i] / dt;
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , m_g2.m_v );
for( size_t i=0 ; i<n ; ++i )
m_xtmp.m_v[i] = x[i] + m_coef.a31 * m_g1.m_v[i] + m_coef.a32 * m_g2.m_v[i];
deriv_func( m_xtmp.m_v , m_dxdtnew.m_v , t + m_coef.c3 * dt );
for( size_t i=0 ; i<n ; ++i )
m_g3.m_v[i] = m_dxdtnew.m_v[i] + dt * m_coef.d3 * m_dfdt.m_v[i] + ( m_coef.c31 * m_g1.m_v[i] + m_coef.c32 * m_g2.m_v[i] ) / dt;
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , m_g3.m_v );
for( size_t i=0 ; i<n ; ++i )
m_xtmp.m_v[i] = x[i] + m_coef.a41 * m_g1.m_v[i] + m_coef.a42 * m_g2.m_v[i] + m_coef.a43 * m_g3.m_v[i];
deriv_func( m_xtmp.m_v , m_dxdtnew.m_v , t + m_coef.c4 * dt );
for( size_t i=0 ; i<n ; ++i )
m_g4.m_v[i] = m_dxdtnew.m_v[i] + dt * m_coef.d4 * m_dfdt.m_v[i] + ( m_coef.c41 * m_g1.m_v[i] + m_coef.c42 * m_g2.m_v[i] + m_coef.c43 * m_g3.m_v[i] ) / dt;
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , m_g4.m_v );
for( size_t i=0 ; i<n ; ++i )
m_xtmp.m_v[i] = x[i] + m_coef.a51 * m_g1.m_v[i] + m_coef.a52 * m_g2.m_v[i] + m_coef.a53 * m_g3.m_v[i] + m_coef.a54 * m_g4.m_v[i];
deriv_func( m_xtmp.m_v , m_dxdtnew.m_v , t + dt );
for( size_t i=0 ; i<n ; ++i )
m_g5.m_v[i] = m_dxdtnew.m_v[i] + ( m_coef.c51 * m_g1.m_v[i] + m_coef.c52 * m_g2.m_v[i] + m_coef.c53 * m_g3.m_v[i] + m_coef.c54 * m_g4.m_v[i] ) / dt;
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , m_g5.m_v );
for( size_t i=0 ; i<n ; ++i )
m_xtmp.m_v[i] += m_g5.m_v[i];
deriv_func( m_xtmp.m_v , m_dxdtnew.m_v , t + dt );
for( size_t i=0 ; i<n ; ++i )
xerr[i] = m_dxdtnew.m_v[i] + ( m_coef.c61 * m_g1.m_v[i] + m_coef.c62 * m_g2.m_v[i] + m_coef.c63 * m_g3.m_v[i] + m_coef.c64 * m_g4.m_v[i] + m_coef.c65 * m_g5.m_v[i] ) / dt;
boost::numeric::ublas::lu_substitute( m_jac.m_v , m_pm.m_v , xerr );
for( size_t i=0 ; i<n ; ++i )
xout[i] = m_xtmp.m_v[i] + xerr[i];
}
