An example of how to use Boost.odeint to integrate a simple ODE.

boost_ode.cpp

#include <array>                     // for std::array
#include <boost/numeric/odeint.hpp>  // for all boost ODE int
#include <functional>                // for std::ref
#include <iostream>                  // for std::cout
#include <vector>                    // for std::vector

static constexpr size_t number_of_dependent_variables = 1;

// The Observer is a class that stores the state of the ODE integration so we
// can get not just the end result of the integration, but a solution over time.
// That is, we have x(t) instead of just x(t_final).
class Observer {
 public:
  void operator()(
      const std::array<double, number_of_dependent_variables>& current_x,
      const double current_time) noexcept {
    x.push_back(current_x[0]);
    time.push_back(current_time);
  }
  std::vector<double> x;
  std::vector<double> time;
};

int main() {
  const double x_start_value = 0.0;
  const double x_end_value = 2.0;
  // The Delta x is somewhat arbitrary. This is basically the size of the
  // rectangles in a Riemann sum. So if you remember doing Riemann sums to
  // estimate an integral by cutting up the area under the curve into a bunch of
  // rectangles, initial_delta_x_for_integration is the width of those
  // rectangles.
  //
  // This also determines how often we store the current solution in the
  // observer object below.
  const double initial_delta_x_for_integration = 0.1;

  // Tolerances means how accurate we want the solution. Relative tolerance is
  // "how many digits" while absolute tolerence is "we only care about the
  // solution if its value is larger than this".
  //
  // Absolute tolerance (safeguards against zeros is the solution):
  // if y_numerical - y_exact < absolute_tolerance:
  //   "things are fine, we don't care about tiny numbers"
  const double absolute_tolerance = 1.0e-8;
  // Relative tolerance:
  // (y_numerical - y_exact) / abs(y_numerical)
  const double relative_tolerance = 1.0e-8;

  // All these super long types are just Boost being.... "flexible" (I'd say
  // difficult, actually)
  using StateDopri5 = boost::numeric::odeint::runge_kutta_dopri5<
      std::array<double, number_of_dependent_variables>>;
  boost::numeric::odeint::dense_output_runge_kutta<
      boost::numeric::odeint::controlled_runge_kutta<StateDopri5>>
      dopri5 = make_dense_output(absolute_tolerance, relative_tolerance,
                                 StateDopri5{});

  // The observer object will store the result at specific times. Which times
  // can be controlled by choosing changing initial_delta_x_for_integration.
  Observer observer_fixed_step_size{};

  // The observer object will store the result at specific times. Which times
  // are specified in the times_to_observe_at std::vector<double>. The
  // integration range is from the first value to the last.
  Observer observer_at_chosen_steps{};
  std::vector<double> times_to_observe_at{
      x_start_value, 0.23, 0.4444, 0.8888, 1.374, 1.843, x_end_value};

  // This is the initial condition and will be updated as we integrate.
  std::array<double, number_of_dependent_variables> x{{1.0}};
  // We want to solve:
  // dx / dt = x

  // Integrate while observing at constant step size
  boost::numeric::odeint::integrate_const(
      dopri5,
      [](const std::array<double, number_of_dependent_variables>&
             current_value_of_x,
         std::array<double, number_of_dependent_variables>&
             current_time_derivative_of_x,
         const double current_time_t) noexcept {
        // Note we don't use the time explicitly!
        (void)current_time_t;
        // This computes the dx/dt
        current_time_derivative_of_x[0] = current_value_of_x[0];
      },
      x, x_start_value, x_end_value, initial_delta_x_for_integration,
      std::ref(observer_fixed_step_size));

  std::cout << "Printing out solution obtained from the fixed step size "
               "observer.\n";
  for (size_t time_index = 0; time_index < observer_fixed_step_size.x.size();
       ++time_index) {
    std::cout << observer_fixed_step_size.time[time_index] << " "
              << observer_fixed_step_size.x[time_index] << "\n";
  }

  // Need to reset to initial condition before integrating again
  x[0] = 1.0;

  // Integrate while observing at specified times
  boost::numeric::odeint::integrate_times(
      dopri5,
      [](const std::array<double, number_of_dependent_variables>&
             current_value_of_x,
         std::array<double, number_of_dependent_variables>&
             current_time_derivative_of_x,
         const double current_time_t) noexcept {
        // Note we don't use the time explicitly!
        (void)current_time_t;
        // This computes the dx/dt
        current_time_derivative_of_x[0] = current_value_of_x[0];
      },
      x, times_to_observe_at.begin(), times_to_observe_at.end(),
      initial_delta_x_for_integration, std::ref(observer_at_chosen_steps));

  std::cout
      << "\n\nPrinting out solution obtained at explicitly chosen times.\n";
  for (size_t time_index = 0; time_index < observer_at_chosen_steps.x.size();
       ++time_index) {
    std::cout << observer_at_chosen_steps.time[time_index] << " "
              << observer_at_chosen_steps.x[time_index] << "\n";
  }
}

Dear Nils hi, I understand little bit your program, but I would like to do a question about the integration of a tabulated data for example I have an integral like to \int_a^b f(x) dx where my f(x) is a tabulated function (x;f(x)) contained at data_function.dat file.
Question:
How I would pass the data_function.dat file to function integrate_const( )?

M.

Hi @mmanyas!

This is a great question. Integration of a table is a bit trickier. The code here chooses the integration step size adaptively to reach a target relative and absolute error tolerance. Naively, with a table you are "stuck" choosing steps that align with the table. Something like an RK4 integrator could be used assuming the data in the table is uniformly spaced. Another option that I've had great success with in the past is creating an interpolant, such as a cubic spline (https://www.boost.org/doc/libs/1_65_0/libs/math/doc/html/math_toolkit/interpolate/cubic_b.html), and then capturing that in the lambda passed to the integrator on line 77:

boost::math::cubic_b_spline<double> spline(table.begin(), table.end(), table_x0, table_delta_x);
  boost::numeric::odeint::integrate_times(
      dopri5,
      [&spline](const std::array<double, number_of_dependent_variables>&
             current_value_of_x,
         std::array<double, number_of_dependent_variables>&
             current_time_derivative_of_x,
         const double current_time_t) noexcept {
        // This computes the dx/dt
        current_time_derivative_of_x[0] = spline(current_time_t);
      },
      x, times_to_observe_at.begin(), times_to_observe_at.end(),
      initial_delta_x_for_integration, std::ref(observer_at_chosen_steps));

Note that this assumes the spline data is the f(x) that you are trying to integrate. I also didn't try to compile the code snippet, so there might be some bugs. You can also use a different interpolant, which will be required if your table is uniformly spaced. If your table is uniformly spaced in say log-space, then you can create the spline in log-space and just exponentiate after.

Hope that helps!

Cheers,
Nils

Hi, very comprehensive example. Thanks for it. I have a question, I'm trying to solve a system of ODEs using boost adaptive rkf78 method, how should I write my observer function to be in order for it to store the values of the various coupled variables at each adaptive timestep the solver takes?

Hi @akg-comp, all you need to do is change std::vector<double> x to std::vector<std::array<double, number_of_dependent_variables>> vars, then do vars.push_back(std::array<double, number_of_dependent_variables>{{current_x[0], current_x[1], current_x[2], ...}}); where ... is the extra vars you have.

Cheers,

Nils