我试图求解一维波动方程,我编写了数值计算解决方案的程序,并制作了动画,将数据保存在文件中。我不知道如何修复错误并最终获得工作代码。
u_tt = a**2 * u_xx + f(x,t)
因此,我附上我的代码(Python 3.9)和错误消息:
import numpy as np
import math
import matplotlib.pyplot as plt
import os
import time
import glob
def sol(I, V, f, a, L, C, T, U_0, U_L, dt, user_func = None):
"""
solver for wave equation
u_tt = a**2*u_xx + f(x,t) (0,L) where u=0 for
x=0,L, for t in (0,T].
:param I:
:param V:
:param f:
:param a:
:param L:
:param C:
:param T:
:param U_0:
:param U_L:
:param dt:
:param user_func:
:return:
"""
nt = int(round(T / dt))
t = np.linspace(0, nt * dt, nt + 1) # array for time points
dx = dt * a / float(C)
nx = int(round(L / dx))
x = np.linspace(0, L, nx + 1) # array for coord points
q = a ** 2
C2 = (dt / dx) ** 2
dt2 = dt * dt
# --- checking f(x,t) ---
if f is None or f == 0:
f = lambda x, t: 0
# --- check the initial conds dU(x,0)/dt ---
if V is None or V == 0:
V = lambda x: 0
# boundary conds
if U_0 is not None:
if isinstance(U_0, (float, int)) and U_0 == 0:
U_0 = lambda t: 0
if U_L is not None:
if isinstance(U_L, (float, int)) and U_L == 0:
U_L = lambda t: 0
# --- allocate memory ---
u = np.zeros(nx + 1)
u_n = np.zeros(nx + 1)
u_nm = np.zeros(nx + 1)
# --- valid indexing check ---
Ix = range(0, nx + 1)
It = range(0, nt + 1)
# --- set the boundary conds ---
for i in range(0, nx + 1):
u_n[i] = I(x[i])
if user_func is not None:
user_func(u_n, x, t, 0)
# --- finite difference step ---
for i in Ix[1:-1]:
u[i] = u_n[i] + dt * V(x[i]) + 0.5 * C2 * (0.5 * (q[i] + q[i + 1]) * (u_n[i + 1] - u_n[i]) -
0.5 * (q[i] + q[i - 1]) * (u_n[i] - u_n[i - 1])) + 0.5 * dt2 * f(x[i], t[0])
i = Ix[0]
if U_0 is None:
# set the boundary conds (x=0: i-1 -> i+1 u[i-1]=u[i+1]
# where du/dn = 0, on x=L: i+1 -> i-1 u[i+1]=u[i-1])
ip1 = i + 1
im1 = ip1 # i-1 -> i+1
u[i] = u_n[i] + dt * V(x[i]) + \
0.5 * C2 * (0.5 * (q[i] + q[ip1]) * (u_n[ip1] - u_n[i]) - 0.5 * (q[i] + q[im1])
* (u_n[i] - u_n[im1])) + 0.5 * dt2 * f(x[i], t[0])
else:
u[i] = U_0(dt)
i = Ix[-1]
if U_L is None:
im1 = i - 1
ip1 = im1 # i+1 -> i-1
u[i] = u_n[i] + dt * V(x[i]) + \
0.5 * C2 * (0.5 * (q[i] + q[ip1]) * (u_n[ip1] - u_n[i]) - 0.5 * (q[i] + q[im1]) * (u_n[i] - u_n[im1])) + \
0.5 * dt2 * f(x[i], t[0])
else:
u[i] = U_L(dt)
if user_func is not None:
user_func(u, x, t, 1)
# update data
u_nm, u_n, u = u_n, u, u_nm
# --- time looping ---
for n in It[1:-1]:
# update all inner points
for i in Ix[1:-1]:
u[i] = - u_nm[i] + 2 * u_n[i] + \
C2 * (0.5 * (q[i] + q[i + 1]) * (u_n[i + 1] - u_n[i]) -
0.5 * (q[i] + q[i - 1]) * (u_n[i] - u_n[i - 1])) + dt2 * f(x[i], t[n])
# --- set boundary conds ---
i = Ix[0]
if U_0 is None:
# set the boundary conds
# x=0: i-1 -> i+1 u[i-1]=u[i+1] where du/dn=0
# x=L: i+1 -> i-1 u[i+1]=u[i-1] where du/dn=0
ip1 = i + 1
im1 = ip1
u[i] = - u_nm[i] + 2 * u_n[i] + \
C2 * (0.5 * (q[i] + q[ip1]) * (u_n[ip1] - u_n[i]) - 0.5 * (q[i] + q[im1]) * (u_n[i] - u_n[im1])) + \
dt2 * f(x[i], t[n])
else:
u[i] = U_0(t[n + 1])
i = Ix[-1]
if U_L is None:
im1 = i - 1
ip1 = im1
u[i] = - u_nm[i] + 2 * u_n[i] + \
C2 * (0.5 * (q[i] + q[ip1]) * (u_n[ip1] - u_n[i]) - 0.5 * (q[i] + q[im1]) * (u_n[i] - u_n[im1])) + \
dt2 * f(x[i], t[n])
else:
u[i] = U_L(t[n + 1])
if user_func is not None:
if user_func(u, x, t, n + 1):
break
u_nm, u_n, u = u_n, u, u_nm
return u, x, t
# --- here function for return functions ---
# return func(x)
def func(x):
"""
:param x:
:return:
"""
return # expression
# start simulate and animate or visualisation and savin the data from file
def simulate(
I, V, f, a, L, C, T, U_0, U_L, dt, # params
umin, umax, # amplitude
animate = True, # animate or not?
solver_func = sol, # call the solver
mode = 'plotter', # mode: plotting the graphic or saving to file
):
# code for visualization and simulate
...........
# start simulate
solver_func(I, V, f, a, L, C, T, U_0, U_L, dt, user_func)
return 0
def task( ):
'''
test tasking for solver and my problem
:return:
'''
I
L = 1
a = 1
C = 0.85
T = 1
dt = 0.05
U_0, U_L, V, f
umax = 2
umin = -umax
simulate(I, V, f, a, L, C, T, U_0, U_L, dt, umax, umin, animate = True, solver_func = sol, mode = 'plotter',)
if __name__ == '__main__':
task()
我得到了同样的错误:
File "C:\\LR2-rep\wave_eq_1d.py", line 102, in sol
u[i] = u_n[i] + dt * V(x[i]) + 0.5 * C2 * (0.5 * (q[i] + q[i + 1]) * (u_n[i + 1] - u_n[i]) -
我理解这个错误的含义,但我不明白如何修复它,而且近两周来我一直无法编写程序。。。我请求帮助解决这个问题!提前非常感谢!
