[英]Solving system of coupled differential equations using Runge-Kutta in python
[英]Weird results obtained while solving a set of coupled differential equations (using a sparse array) in python
在尝试求解耦合微分方程系统并重现附图中显示的结果时,我尝试了一周都无济于事。 我似乎也得到了奇怪的结果,如图所示。 我似乎不知道我可能做错了什么。耦合微分方程组是使用 Newman 的 BAND 求解的。 这是 python 实现的链接: python solution using BAND 。 如果附件不够清晰,还有另一个指向问题原始图像的链接:在这里你可以找到更清晰的问题图像。 现在我要做的是通过使用 sympy 和 numpy 的组合直接从离散化方程创建一个稀疏数组来解决相同的问题,然后使用 scipy 的 spsolve 求解。 下面是我的代码。 我需要一些帮助来弄清楚我做错了什么。 我在我的代码中将变量表示为 c1 = cA、c2 = cB、c3 = cC、c4 = cD。 等式 2 已经线性化,phi10 和 phi20 是变量 cC 和 cD 的试验值。
# import modules
import numpy as np
import sympy
from sympy.core.function import _mexpand
import scipy as sp
import scipy.sparse as ss
import scipy.sparse.linalg as ssl
import matplotlib.pyplot as plt
# define functions
def flatten(t):
"""
function to flatten lists
"""
return [item for sublist in t for item in sublist]
def get_coeffs(coeff_dict, func_vars):
"""
function to extract coefficients from variables
and form the sparse symbolic array
"""
c = coeff_dict
for i in list(c.keys()):
b, _ = i.as_base_exp()
if b == i:
continue
if b in c:
c[i] = 0
if any(k.has(b) for k in c):
c[i] = 0
return [coeff_dict[val] for val in func_vars]
# Constants for the problem
I = 0.1 # A/cm2
L = 1.0 # distance (x) in cm
m = 100 # grid spacing
h = L / (m-1)
a = 23300 # 1/cm
io = 2e-7 # A/cm2
n = 1
F = 96500 # C/mol
R = 8.314 # J/mol-K
T = 298 # K
sigma = 20 # S/cm
kappa = 0.06 # S/cm
alpha = 0.5
beta = -(1-alpha)*n*F/R/T
phi10 , phi20 = 5, 0.5 # these are just guesses
P = a*io*np.exp(beta*(phi10-phi20))
j = sympy.symbols('j',integer = True)
cA = sympy.IndexedBase('cA')
cB = sympy.IndexedBase('cB')
cC = sympy.IndexedBase('cC')
cD = sympy.IndexedBase('cD')
# write the boundary conditions at x = 0
bc=[cA[1], cB[1],
(4/3) * cC[2] - (1/3)*cC[3], # use a three point approximation for cC_prime
cD[1]
]
# form a list of expressions from the boundary conditions and equations
expr=flatten([bc,flatten([[
-cA[j-1] - cB[j-1] + cA[j+1] + cB[j+1],
cB[j-1] - 2*h*P*beta*cC[j] + 2*h*P*beta*cD[j] - cB[j+1],
-sigma*cC[j-1] + 2*h*cA[j] + sigma * cC[j+1],
-kappa * cD[j-1] + 2*h * cB[j] + kappa * cD[j+1]] for j in range(2, m)])])
vars = [cA[j], cB[j], cC[j], cD[j]]
# flatten the list of variables
unknowns = flatten([[cA[j], cB[j], cC[j], cD[j]] for j in range(1,m)])
var_len = len(unknowns)
# # # substitute in the boundary conditions at x = L while getting the coefficients
A = sympy.SparseMatrix([get_coeffs(_mexpand(i.subs({cA[m]:I}))\
.as_coefficients_dict(), unknowns) for i in expr])
# convert to a numpy array
mat_temp = np.array(A).astype(np.float64)
# you can view the sparse array with this
fig = plt.figure(figsize=(6,6))
ax = fig.add_axes([0,0, 1,1])
cmap = plt.cm.binary
plt.spy(mat_temp, cmap = cmap, alpha = 0.8)
def solve_sparse(b0, error):
# create the b column vector
b = np.copy(b0)
b[0:4] = np.array([0.0, I, 0.0, 0.0])
b[var_len-4] = I
b[var_len-3] = 0
b[var_len-2] = 0
b[var_len-1] = 0
print(b.shape)
old = np.copy(b0)
mat = np.copy(mat_temp)
b_2 = np.copy(b)
resid = 10
lss = 0
while lss < 100:
mat_2 = np.copy(mat)
for j in range(3, var_len - 3, 4):
# update the forcing term of equation 2
b_2[j+2] = 2*h*(1-beta*old[j+3]+beta*old[j+4])*a*io*np.exp(beta*(old[j+3]-old[j+4]))
# update the sparse array at every iteration for variables cC and cD in equation2
mat_2[j+2, j+3] += 2*h*beta*a*io*np.exp(beta*(old[j+3]-old[j+4]))
mat_2[j+2, j+4] += 2*h*beta*a*io*np.exp(beta*(old[j+3]-old[j+4]))
# form the column sparse matrix
A_s = ss.csc_matrix(mat_2)
new = ssl.spsolve(A_s, b_2).flatten()
resid = np.sum((new - old)**2)/var_len
lss += 1
old = np.copy(new)
return new
val0 = np.array([[0.0, 0.0, 0.0, 0.0] for _ in range(m-1)]).flatten() # form an array of initial values
error = 1e-7
## Run the code
conc = solve_sparse(val0, error).reshape(m-1, len(vars))
conc.shape # gives (99, 4)
# Plot result for cA:
plt.plot(conc[:,0], marker = 'o', linestyle = '')
发生的事情似乎很清楚。 您正在使用中心欧拉方法作为离散化,对于u'=F(u)
这读作
u[j+1]-u[j-1] = 2*h*F(u[j])
这种方法仅是弱稳定的,并且允许奇数和偶数索引的子序列相当独立地演化。 作为方程,这意味着该解可能近似系统ue'=F(uo), uo'=F(ue)
具有独立函数ue, uo
遵循偶数或奇数子序列的路径。
为了避免这种情况,需要非常小心地处理边界点的边界值和方程。
但是使用梯形方法可以避免所有这些不愉快
u[j+1]-u[j] = 0.5*h*(F(u[j+1])+F(u[j]))
这也减少了系统矩阵的带宽。
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