Question

HW9-9 Write a python script to solve the Laplace equation in two dimensions.

code that I started:

# THIS IS NEEDED TO USE ARRAYS IN PYTHON:
from array import *
# THIS IS NEEDED TO MAKE PLOTS IN PYTHON:
import matplotlib.pyplot as plt
# IMPORT A ROUTINE FROM NUMPY USEFUL FOR CREATING AN
# ARRAY FILLED WITH ZEROS
from numpy import zeros

# THIS COMMAND CREATES AN ARRAY OF ZEROS WITH DESIRED SHAPE:
V=zeros([31,21])
newV=zeros([31,21])

# SET UP THE BOUNDARY VALUES: V=8 ON TOP OF BOX
iy=20
for ix in range(0,31):
V[ix,iy]=8.0

# DO THE ITERATIONS (MAY WANT TO ADJUST Nt)
Nt=100
for it in range(0,Nt):
for iy in range(1,20):
for ix in range(1,30):

.... I dont know how to finish ....

**** please help me complete the script please

Answer 1

I'm providing two solutions :-

#1

from visual import*
from visual.graph import*
import numpy as np

 x = y = 7
del = 2

vtop = [-1,-.67,-.33,.00,.33,.67,1]
vbottom = [-1,-.67,-.33,.00,.33,.67,1]
vleft = -1
vright = 1

vguess= 0

x,y = np.meshgrid(np.arange(0,lenx), np.arange(0,leny))

v = np.empty((x,y))
v.fill(vguess)

v[(y-1):,:] = vtop
v [:1,:] = vbottom
v[:,(x-1):] = vright
v[:,:1] = vleft

boundary = 500

for iteration in range (0,boundary):
    for i in range(1,x):
        for j in range(1,y-1):
            v[i,j] = 

                      v[i,j] = .25*(v[i+1][j] + v[i-1][j] + v[i][j+1] + v[i][j-1])

            print (v)

#2

import matplotlib
import matplotlib.pyplot as plt
import numpy as np
import time
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm


plt.ion()
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.set_zlim(-1.01, 1.01)


def draw_plot(x, y, u):
    ax.clear()
    ax.set_zlim(-1.01, 1.01)
    ax.plot_surface(x, y, u, rstride=1, cstride=1, cmap=cm.coolwarm,
                    linewidth=0, antialiased=True)
    plt.pause(1e-5)


# Create  grid
m = 21
mesh_range = np.arange(-1, 1, 2/(m-1))
x, y = np.meshgrid(mesh_range, mesh_range)

# base case
u = np.exp(-5 * (x**2 + y**2))

draw_plot(x, y, u)

n = list(range(1, m-1)) + [m-2]
e = n
s = [0] + list(range(0, m-2))
w = s


def pde_step(u):
    """ PDE calculation at a single time step t  """
    return (u[n, :]+u[:, e]+u[s, :]+u[:, w])/4.


k = 5
U_step = u

for it in range(500):
    U_step = pde_step(U_step)

    # Every k steps, draw the graphics
    if it % k == 0:
        draw_plot(x, y, U_step)

while True:
    plt.pause(1e-5)