boost odeint 给出了与 Python3 scipy 截然不同的值

Question

I am trying to integrate a very simple ODE using boost odeint. For some cases, the values are the same (or very similar) to Python's scipy odeint function. But for other initial conditions, the values are vastly different.

The function is: d(uhat) / dt = - alpha^2 * kappa^2 * uhat where alpha is 1.0, and kappa is a constant depending on the case (see values below).

I have tried several different ODE solvers from boost, and none seem to work.

Update: The code below is now working.

In the code below, the first case gives nearly identical results, the 2nd case is kind of trivial (but reassuring), and the 3rd case gives erroneous answers in the C++ version.

Here is the C++ version:

#include <boost/numeric/odeint.hpp>
#include <cstdlib>
#include <iostream>

typedef boost::numeric::odeint::runge_kutta_dopri5<double> Stepper_Type;

struct ResultsObserver
{
    std::ostream& m_out;
    ResultsObserver( std::ostream &out ) : m_out( out ) { }
    void operator()(const State_Type& x , double t ) const
    {
      m_out << t << "  :  " << x << std::endl;
    }
};

// The rhs:  d_uhat_dt = - alpha^2 * kappa^2 * uhat
class Eq {
public:
  Eq(double alpha, double kappa)
    : m_constant(-1.0 * alpha * alpha * kappa * kappa) {}
  
  void operator()(double uhat, double& d_uhat_dt, const double t) const
  {
    d_uhat_dt = m_constant * uhat;
  }

private:
  double m_constant;
};

void integrate(double kappa, double initValue)
{
  const unsigned numTimeIncrements = 100;
  const double dt = 0.1;
  const double alpha = 1.0;

  double uhat = initValue;      //Init condition
  std::vector<double> uhats;    //Results vector

  Eq rhs(alpha, kappa);         //The RHS of the ODE
  
  //This is what I was doing that did not work
  //
  //boost::numeric::odeint::runge_kutta_dopri5<double> stepper;
  //for(unsigned step = 0; step < numTimeIncrements; ++step) {
  //  uhats.push_back(uhat);
  //  stepper.do_step(rhs, uhat, step*dt, dt);
  //}

  //This works
  integrate_const(
     boost::numeric::odeint::make_dense_output<Stepper_Type>( 1E-12, 1E-6 ),
     rhs, uhat, startTime, endTime, dt, ResultsObserver(std::cout) 
  );

  std::cout << "kappa = " << kappa << ", initial value = " << initValue << std::endl;
  for(auto val : uhats)
    std::cout << val << std::endl;
  std::cout << "---" << std::endl << std::endl;
}

int main() {

  const double kappa1 = 0.062831853071796;
  const double initValue1 = -187.097241230045967;
  integrate(kappa1, initValue1);

  const double kappa2 = 28.274333882308138;
  const double initValue2 = 0.000000000000;
  integrate(kappa2, initValue2);

  const double kappa3 = 28.337165735379934;
  const double initValue3 = -0.091204068895190;
  integrate(kappa3, initValue3);
  
  return EXIT_SUCCESS;
}

and the corresponding Python3 version:

enter code here
#!/usr/bin/env python3

import numpy as np
from scipy.integrate import odeint

def Eq(uhat, t, kappa, a):
    d_uhat = -a**2 * kappa**2 * uhat
    return d_uhat

def integrate(kappa, initValue):
    dt = 0.1
    t = np.arange(0,10,dt)
    a = 1.0
    print("kappa = " + str(kappa))
    print("initValue = " + str(initValue))
    uhats = odeint(Eq, initValue, t, args=(kappa,a))
    print(uhats)
    print("---")
    print()

kappa1 = 0.062831853071796
initValue1 = -187.097241230045967
integrate(kappa1, initValue1)

kappa2 = 28.274333882308138
initValue2 = 0.000000000000
integrate(kappa2, initValue2)

kappa3 = 28.337165735379934
initValue3 = -0.091204068895190
integrate(kappa3, initValue3)

Answer 1

With boost::odeint you should use the integrate... interface functions.

What happens in your code using do_step is that you use the dopri5 method as a fixed-step method with your provided step size. In connection with the large coefficient L=kappa^2 of about 800, the product L*dt=80 is far outside the stability region of the method, the boundary is between values of 2 to 3.5. Divergence or oscillating divergence is the expected result.

What should happen and what is implemented in the scipy odeint, ode-dopri5 or solve_ivp-RK45 methods is a method with adaptive step size. Internally the optimal step size for the error tolerances is chosen, and in each internal step an interpolation polynomial is determined. The output or observed values are computed with this interpolator, also called "dense output" if the interpolation function is returned from the integrator. One result of the error control is that the step size is always inside the stability region, for stiff problems with medium error tolerances on or close to the boundary of this region.

Answer 2

This has all the hallmarks of precision issues.

Simply replacing double with long double gives:

Live On Compiler Explorer

#include <boost/numeric/odeint.hpp>
#include <boost/multiprecision/cpp_bin_float.hpp>
#include <boost/multiprecision/cpp_dec_float.hpp>
#include <fmt/ranges.h>
#include <fmt/ostream.h>
#include <iostream>

using Value = long double;
//using Value = boost::multiprecision::number<
    //boost::multiprecision::backends::cpp_bin_float<100>,
    //boost::multiprecision::et_off>;

// The rhs:  d_uhat_dt = - alpha^2 * kappa^2 * uhat
class Eq {
  public:
    Eq(Value alpha, Value kappa)
        : m_constant(-1.0 * alpha * alpha * kappa * kappa)
    {
    }

    void operator()(Value uhat, Value& d_uhat_dt, const Value) const
    {
        d_uhat_dt = m_constant * uhat;
    }

  private:
    Value m_constant;
};

void integrate(Value const kappa, Value const initValue)
{
    const unsigned numTimeIncrements = 100;
    const Value    dt                = 0.1;
    const Value    alpha             = 1.0;

    Value              uhat = initValue; // Init condition
    std::vector<Value> uhats;            // Results vector

    Eq rhs(alpha, kappa); // The RHS of the ODE
    boost::numeric::odeint::runge_kutta_dopri5<Value> stepper;

    for (unsigned step = 0; step < numTimeIncrements; ++step) {
        uhats.push_back(uhat);
        auto&& stepdt = step * dt;
        stepper.do_step(rhs, uhat, stepdt, dt);
    }

    fmt::print("kappa = {}, initial value = {}\n{}\n---\n", kappa, initValue,
               uhats);
}

int main() {
    for (auto [kappa, initValue] :
         {
             std::pair //
              { 0.062831853071796 , -187.097241230045967 },
              {28.274333882308138 ,    0.000000000000    },
              {28.337165735379934 ,   -0.091204068895190 },
         }) //
    {
        integrate(kappa, initValue);
    }
}

Prints

kappa = 0.0628319, initial value = -187.097
[-187.097, -187.023, -186.95, -186.876, -186.802, -186.728, -186.655, -186.581, -186.507, -186.434, -186.36, -186.287, -186.213, -186.139, -186.066, -185.993, -185.919, -185.846, -185.772, -185.699, -185.626, -185.553, -185.479, -185.406, -185.333, -185.26, -185.187, -185.114, -185.04, -184.967, -184.894, -184.821, -184.748, -184.676, -184.603, -184.53, -184.457, -184.384, -184.311, -184.239, -184.166, -184.093, -184.021, -183.948, -183.875, -183.803, -183.73, -183.658, -183.585, -183.513, -183.44, -183.368, -183.296, -183.223, -183.151, -183.079, -183.006, -182.934, -182.862, -182.79, -182.718, -182.645, -182.573, -182.501, -182.429, -182.357, -182.285, -182.213, -182.141, -182.069, -181.998, -181.926, -181.854, -181.782, -181.71, -181.639, -181.567, -181.495, -181.424, -181.352, -181.281, -181.209, -181.137, -181.066, -180.994, -180.923, -180.852, -180.78, -180.709, -180.638, -180.566, -180.495, -180.424, -180.353, -180.281, -180.21, -180.139, -180.068, -179.997, -179.926]
---
kappa = 28.2743, initial value = 0
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
---
kappa = 28.3372, initial value = -0.0912041
[-0.0912041, -38364100, -1.61375e+16, -6.78809e+24, -2.85534e+33, -1.20107e+42, -5.0522e+50, -2.12516e+59, -8.93928e+67, -3.76022e+76, -1.5817e+85, -6.65327e+93, -2.79864e+102, -1.17722e+111, -4.95186e+119, -2.08295e+128, -8.76174e+136, -3.68554e+145, -1.55029e+154, -6.52114e+162, -2.74306e+171, -1.15384e+180, -4.85352e+188, -2.04159e+197, -8.58774e+205, -3.61235e+214, -1.5195e+223, -6.39163e+231, -2.68858e+240, -1.13092e+249, -4.75713e+257, -2.00104e+266, -8.41718e+274, -3.54061e+283, -1.48932e+292, -6.26469e+300, -2.63518e+309, -1.10846e+318, -4.66265e+326, -1.9613e+335, -8.25002e+343, -3.47029e+352, -1.45975e+361, -6.14028e+369, -2.58285e+378, -1.08645e+387, -4.57005e+395, -1.92235e+404, -8.08618e+412, -3.40137e+421, -1.43075e+430, -6.01833e+438, -2.53155e+447, -1.06487e+456, -4.47929e+464, -1.88417e+473, -7.92559e+481, -3.33382e+490, -1.40234e+499, -5.89881e+507, -2.48128e+516, -1.04373e+525, -4.39033e+533, -1.84675e+542, -7.76818e+550, -3.26761e+559, -1.37449e+568, -5.78166e+576, -2.432e+585, -1.023e+594, -4.30314e+602, -1.81008e+611, -7.61391e+619, -3.20272e+628, -1.34719e+637, -5.66684e+645, -2.3837e+654, -1.00268e+663, -4.21768e+671, -1.77413e+680, -7.4627e+688, -3.13911e+697, -1.32044e+706, -5.55429e+714, -2.33636e+723, -9.82768e+731, -4.13392e+740, -1.73889e+749, -7.31449e+757, -3.07677e+766, -1.29421e+775, -5.44399e+783, -2.28996e+792, -9.6325e+800, -4.05182e+809, -1.70436e+818, -7.16922e+826, -3.01567e+835, -1.26851e+844, -5.33587e+852]
---

As you can see, I tried some simple takes to get it to use Boost Multiprecision, but that didn't immediately work and may require someone with actual understanding of the maths / IDEINT.