如何在 scipy 集成模块“solve_ivp”中正确设置“rtol”和“atol”以求解具有未知解析解的 ODE 系统？

Question

I was trying to reproduce some results of ode45 solver in Python using solve_ivp. Though all parameters, initial conditions, step size, and 'atol' and 'rtol' (which are 1e-6 and 1e-3) are same, I am getting different solutions. Both of the solutions are converging to a periodic solution but of different kind. As solve_ivp uses same rk4(5) method as ode45, this discrepancy in the final result is not quite understable. How can we know which one is the correct solution? The code is included below

import sys
import numpy as np
from scipy.integrate import solve_ivp
#from scipy import integrate
import matplotlib.pyplot as plt
from matplotlib.patches import Circle


# Pendulum rod lengths (m), bob masses (kg).

L1, L2, mu, a1 = 1, 1, 1/5, 1
m1, m2, B = 1, 1, 0.1
# The gravitational acceleration (m.s-2).

g = 9.81

# The forcing frequency,forcing amplitude

w, a_m =10, 4.5
A=(a_m*w**2)/g
A1=a_m/g

def deriv(t, y, mu, a1, B, w, A): # beware of the order of the aruments
"""Return the first derivatives of y = theta1, z1, theta2, z2, z3."""
a, c, b, d, e = y

#c, s = np.cos(theta1-theta2), np.sin(theta1-theta2)
adot = c
cdot = (-(1-A*np.sin(e))*(((1+mu)*np.sin(a))-(mu*np.cos(a-b)*np.sin(b)))-((mu/a1)*((d**2)+(a1*np.cos(a-b)*c**2))*np.sin(a-b))-(2*B*(1+(np.sin(a-b))**2)*c)-((2*B*A/w)*(2*np.sin(a)-(np.cos(a-b)*np.sin(b)))*np.cos(e)))/(1+mu*(np.sin(a-b))**2)

bdot = d
ddot = ((-a1*(1+mu)*(1-A*np.sin(e))*(np.sin(b)-(np.cos(a-b)*np.sin(a))))+(((a1*(1+mu)*c**2)+(mu*np.cos(a-b)*d**2))*np.sin(a-b))-((2*B/mu)*(((1+mu*(np.sin(a-b))**2)*d)+(a1*(1-mu)*np.cos(a-b)*c)))-((2*B*a1*A/(w*mu))*(((1+mu)*np.sin(b))-(2*mu*np.cos(a-b)*np.sin(a)))*np.cos(e)))/(1+mu*(np.sin(a-b))**2)
edot = w
return adot, cdot, bdot, ddot, edot





# Initial conditions: theta1, dtheta1/dt, theta2, dtheta2/dt.

y0 = np.array([3.15, -0.1, 3.13, 0.1, 0])


# Do the numerical integration of the equations of motion



sol = integrate.solve_ivp(deriv,[0,40000], y0, args=(mu, a1, B, w, A), method='RK45',t_eval=np.arange(0, 40000, 0.005), dense_output=True, rtol=1e-3, atol=1e-6)


T = sol.t
Y = sol.y

I am expecting similar result from ode45 in MATLAB and solve_ivp in Python. How can I exactly reproduce the result from ode45 in python? What is the reason of discrepancy?

Answer 1

Even if ode45 and RK45 use the same underlying scheme, they do not necessarily use the same exact strategy regarding the evolution of the time step and its adaptation to match the error tolerance. Thus, it is difficult to know which one is better. The only thing you could is simply trying lower tolerances, eg 1e-10. Then, both solutions should end up being virtually identical... Here, your current error tolerance might be insufficiently low, so that small discrepancies in the fine details of both algorithms create a visible difference in the solution.