def Exercise1():
x_subintervals, t_subintervals = 6, 10
x_interval, t_final = [0, 1], .4
init_conditions = 2. * np.maximum(
1. / 5 - np.abs(np.linspace(0, 1, x_subintervals + 1) - 1. / 2), 0.)
x, u = heatexplicit(init_conditions,
x_subintervals,
t_subintervals,
x_interval,
T=t_final,
flag3d="on",
nu=.05)
# # Generates the figures in the heat flow lab
# view = [-.1, 1.1,-.1, .5]
# plt.plot(x,u[:,0],'-k',linewidth=2.0) # Initial Conditions
# plt.plot(x,u[:,0],'ok',linewidth=2.0)
# plt.axis(view)
# plt.savefig('heatexercise1a.pdf')
# plt.clf()
#
# plt.plot(x,u[:,-1],'-k',linewidth=2.0) # State at t = .2
# plt.plot(x,u[:,-1],'ok',linewidth=2.0)
# plt.axis(view)
# plt.savefig('heatexercise1b.pdf')
# plt.clf()
Data = x, u[:, ::1]
Data = Data[0], Data[1].T[0:-1, :]
time_steps = Data[1].shape[0]
interval_length = 50
view = [-.1, 1.1, -.1, .5]
math_animation(Data, time_steps, view, interval_length)
def Exercise1():
x_subintervals, t_subintervals = 6,10
x_interval,t_final = [0,1], .4
init_conditions = 2.*np.maximum(1./5 - np.abs(np.linspace(0,1,x_subintervals+1)-1./2),0.)
x,u = heatexplicit(init_conditions,x_subintervals,t_subintervals,x_interval,T=t_final,flag3d="on",nu=.05)
# # Generates the figures in the heat flow lab
# view = [-.1, 1.1,-.1, .5]
# plt.plot(x,u[:,0],'-k',linewidth=2.0) # Initial Conditions
# plt.plot(x,u[:,0],'ok',linewidth=2.0)
# plt.axis(view)
# plt.savefig('heatexercise1a.pdf')
# plt.clf()
#
# plt.plot(x,u[:,-1],'-k',linewidth=2.0) # State at t = .2
# plt.plot(x,u[:,-1],'ok',linewidth=2.0)
# plt.axis(view)
# plt.savefig('heatexercise1b.pdf')
# plt.clf()
Data = x,u[:,::1]
Data = Data[0], Data[1].T[0:-1,:]
time_steps = Data[1].shape[0]
interval_length=50
view = [-.1, 1.1,-.1, .5]
math_animation(Data,time_steps,view,interval_length)
def Exercise2():
x_subintervals, t_subintervals = 140., 70.
x_interval, t_final = [-12, 12], 1.
init_conditions = np.maximum(
1. - np.linspace(-12, 12, x_subintervals + 1)**2, 0.)
x, u = heatexplicit(init_conditions,
x_subintervals,
t_subintervals,
x_interval,
T=t_final,
flag3d="off",
nu=1.)
# # Generates a figure in the heat flow lab
# view = [-12, 12,-.1, 1.1]
# plt.plot(x,u[:,0],'-k',linewidth=2.0,label="Initial State")
# plt.plot(x,u[:,-1],'--k',linewidth=2.0,label="State at time $t=1$.") # State at t = .2
# plt.xlabel("x",fontsize=16)
# plt.legend(loc=1)
# plt.axis(view)
# plt.savefig('heatexercise2.pdf')
# plt.show()
# plt.clf()
# Animation of results
view = [-12, 12, -.1, 1.1]
Data = x, u[:, ::2]
Data = Data[0], Data[1].T[0:-1, :]
time_steps = Data[1].shape[0]
interval_length = 30
math_animation(Data, time_steps, view, interval_length)
def Exercise2():
x_subintervals, t_subintervals = 140.,70.
x_interval,t_final = [-12,12], 1.
init_conditions = np.maximum(1.-np.linspace(-12,12,x_subintervals+1)**2,0.)
x,u = heatexplicit(init_conditions,x_subintervals,t_subintervals,x_interval,T=t_final,flag3d="off",nu=1.)
# # Generates a figure in the heat flow lab
# view = [-12, 12,-.1, 1.1]
# plt.plot(x,u[:,0],'-k',linewidth=2.0,label="Initial State")
# plt.plot(x,u[:,-1],'--k',linewidth=2.0,label="State at time $t=1$.") # State at t = .2
# plt.xlabel("x",fontsize=16)
# plt.legend(loc=1)
# plt.axis(view)
# plt.savefig('heatexercise2.pdf')
# plt.show()
# plt.clf()
# Animation of results
view = [-12, 12,-.1, 1.1]
Data = x,u[:,::2]
Data = Data[0], Data[1].T[0:-1,:]
time_steps = Data[1].shape[0]
interval_length=30
math_animation(Data,time_steps,view,interval_length)
return
def animate_heat1d(): Domain = 10. Data = heatexplicit(J=320,nu=1.,L=Domain,T=1.,flag3d="animation") Data = Data[0], Data[1].T[0:-1,:] time_steps, view = Data[1].shape[0], [-Domain, Domain,-.1, 1.5] math_animation(Data,time_steps,view) return
def plot3d():
T, L, J, nu = 1., 10., 320, 1.
flag3d = "3d_plot"
X,U = heatexplicit(J,nu,L,T,flag3d)
N = U.shape[1]-1
# #Produce 3D plot of solution
xv,tv = np.meshgrid(np.linspace(-L,L,J/4+1), np.linspace(0,T,N/8+1))
Z = U[::4,::8].T
fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(tv, xv, Z, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
ax.set_xlabel('Y'); ax.set_ylabel('X'); ax.set_zlabel('Z')
plt.show()
return
def plot_error():
# Compare solutions for various discretizations
X, U, L = [], [], 10.
List, LineStyles = [40,80,160,320,640,1280,2560], ['--k','-.k',':k','-k','-k','-k']
for points in List:
x,u = heatexplicit(J=points,nu=1.,L=10.,T=1.,flag3d="off")
X.append(x);
U.append(u[:,-1])
del x, u
print points
# # Plot of solutions for varying discretizations
# for j in range(0,len(LineStyles)): plt.plot(X[j],U[j],LineStyles[j],linewidth=1)
# plt.axis([-10,10,-.1,.4])
# plt.show()
# plt.clf()
h = [2.*L/item for item in List[:-1]]
def graph(X,U,List,h):
# # Error Chart 1: Plot approximate error for each solution
# for j in range(0,len(List)-1): plt.plot(X[j],abs(U[-1][::2**(6-j)]-U[j]),'-k',linewidth=1.2)
for j in range(0,len(List)-1): plt.plot(X[j],abs(U[-1][::List[-1]/List[j]]-U[j]),'-k',linewidth=1.2)
plt.savefig("ApproximateError.pdf")
plt.clf()
# # Error Chart 2: Create a log-log plot of the max error of each solution
# MaxError = [max(abs(U[-1][::2**(6-j)]-U[j] )) for j in range(0,len(List)-1)]
MaxError = [max(abs(U[-1][::List[-1]/List[j]]-U[j] )) for j in range(0,len(List)-1)]
plt.loglog(h,MaxError,'-k',h,MaxError,'ko')
plt.ylabel("Max Error")
plt.xlabel('$\Delta x$')
h, MaxError = np.log(h)/np.log(10),np.log(MaxError)/np.log(10)
print '\n'*2, "Approximate order of accuracy = ", (MaxError[-1]-MaxError[0])/(h[-1]-h[0])
plt.savefig("MaximumError.pdf")
# plt.show()
return
graph(X,U,List,h)
return
