Solving set of Boundary Value Problems

问题

I am trying to solve a set of boundary value problems given by 4 differential equations. I am using bvp_solver in python, and I am getting errors which state 'invalid value encountered in division'. I am assuming this means I am dividing by NaN or 0 at some point, but I am unsure where.

import numpy as np
from scipy.integrate import solve_bvp
import matplotlib.pyplot as plt
%matplotlib inline
alpha = 1
zeta = 1
C_k = 1
sigma = 1
Q = 30
U_0 = 0.1
gamma = 5/3
theta = 3
m = 1.5
def fun(x, y):
    U, dU, B, dB, T, dT, M, dM = y;
    d2U = -2*U_0*Q**2*(1/np.cosh(Q*x))**2*np.tanh(Q*x)-((alpha)/(C_k*sigma))*dB;   
    d2B = -(1/(C_k*zeta))*dU;
    d2T = (1/gamma - 1)*(sigma*dU**2 + zeta*alpha*dB**2);
    d2M = -(dM/T)*dT + (dM/T)*theta*(m+1) - (alpha/T)*B*dB
    return dU, d2U, dB, d2B, dT, d2T, dM, d2M 

def bc(ya, yb):

    return ya[0]+U_0*np.tanh(Q*0.5), yb[0]-U_0*np.tanh(Q*0.5), ya[2]-0, yb[2]-0, ya[4] - 1, yb[4] - 4, ya[6], yb[6] - 1

x = np.linspace(-0.5, 0.5, 500)
y = np.zeros((8, x.size))


sol = solve_bvp(fun, bc, x, y)

If I remove the last two equations for M and dM, then the solution works fine. I have had trouble in the past understanding the return arrays of bvp_solver, but I am confident I understand that now. But I continue to get errors each time I add more equations. Any help is greatly appreciated.

回答1:

Of course this will fail in the first step. You initialize everything to zero, and then in the derivatives function, you divide by T, which is zero from the initialization.

Find a more realistic initialization of T, for instance

x = np.linspace(-0.5, 0.5, 15)
y = np.zeros((8, x.size))
y[4] = 2.5+3*x
y[5] = 3+0*x

desingularize the division, which usually is done similar to

d2M = (-dM*dT + dM*theta*(m+1) - alpha*B*dB) * T/(1e-12+T*T)

It makes always sense to print after sol = solve_bvp(...) the error message print(sol.message). Now that there are more than a few components, I changed the construction of the output tableau to the systematic

%matplotlib inline
plt.figure(figsize=(10,2.5*4))
for k in range(4):
    v,c = ['U','B','T','M'][k],['-+b','-*r','-xg','-om'][k];
    plt.subplot(4,2,2*k+1); plt.plot(sol.x,sol.y[2*k  ],c, ms=2); plt.grid(); plt.legend(["$%c$"%v]);
    plt.subplot(4,2,2*k+2); plt.plot(sol.x,sol.y[2*k+1],c, ms=2); plt.grid(); plt.legend(["$%c'$"%v]);
plt.tight_layout(); plt.savefig("/tmp/bvp3.png"); plt.show(); plt.close()

来源：https://stackoverflow.com/questions/60453812/solving-set-of-boundary-value-problems

标签

python

numpy

ode

differential-equations

boundary