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 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)
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 Burgers(): def bump(x): sigma, mu = 1., 0. gaussian = (1./(sigma*np.sqrt(2*np.pi) ) )*np.exp((-1./2.)*((x-mu)/sigma)**2.) return gaussian*(3.5*np.sin(3*x)+ 3.5) def pde_matrix_func(Y2,Y1): out = np.zeros(len(Y1)) c1 = (theta*delta_t)/(2.*delta_x) c2 = (theta*delta_t)/(delta_x**2.) c3 = ((1.-theta)*delta_t)/(2.*delta_x) c4 = ((1.-theta)*delta_t)/(delta_x**2.) out[1:-1] = ( Y2[1:-1] + c1*(Y2[1:-1] - s)*(Y2[2:] - Y2[:-2]) - c2*(Y2[2:] - 2.*Y2[1:-1] + Y2[:-2])- Y1[1:-1] + c3*(Y1[1:-1] - s)*(Y1[2:] - Y1[:-2]) - c4*(Y1[2:] - 2.*Y1[1:-1] + Y1[:-2]) ) out[0] = Y2[0]-Y1[0] out[-1] = Y2[-1]-Y1[-1] # for j in range(len(Y1)-1,len(Y1)): out[j] = Y2[j]-Y1[j] return out L, T = 20., 1. u_m,u_p = 5.,1. s, a = (u_m + u_p)/2., (u_m - u_p)/2. n, time_steps = 150, 350 delta_x, delta_t = 2.*L/n, T/(n) x = np.linspace(-L,L,n) wave, perturb = s - a*np.tanh((a/2.)*x), bump(x) theta = 1/2. U = np.zeros( (time_steps, n) ) U[0,:] = wave+perturb U[0,0], U[0,-1] = u_m, u_p for j in xrange(1, time_steps): U[j,:] = fsolve(lambda X,Y= U[j-1,:]: pde_matrix_func(X,Y), U[j-1,:] ) print j view = [-L,L,u_p-1,u_m+1] stride = 1 Data = x, U[::stride,:], wave time_steps, wait = int(time_steps/stride), 30 math_animation(Data,time_steps,view,wait)
def example5(): L, T = 1., 2 N_x, N_t = 200, 440 def f(x): return 4.*np.sin(4*np.pi*x) def g(x): return np.zeros(x.shape) view = [0-.1,L+.1,-5,5] Data = wave_1d(f,g,L,N_x,T,N_t,view) stride = 2 Data = Data[0], Data[1][::stride,:] time_steps, wait = int(N_t/stride), 200 math_animation(Data,time_steps,view,wait)
def example2(): L, T = 1., 2 N_x, N_t = 200, 440 m, u_0 = 20, .2 def f(x): return u_0*np.exp(-m**2.*(x-1./2.)**2.) def g(x): return -m**2.*2.*(x-1./2.)*f(x) view = [0-.1,L+.1,-.5,.5] Data = wave_1d(f,g,L,N_x,T,N_t,view) stride = 1 Data = Data[0], Data[1][::stride,:] time_steps, wait = int(N_t/stride), 4 math_animation(Data,time_steps,view,wait)
def example4(): L, T = 1., 2 N_x, N_t = 200, 440 def f(x): y = np.zeros_like(x) y[np.where( (5.*L/11. < x) & (x < 6*L/11. ) ) ] = 1./3 return y def g(x): return np.zeros(x.shape) view = [-.1,L+.1,-.5,.5] Data = wave_1d(f,g,L,N_x,T,N_t,view) stride = 2 Data = Data[0], Data[1][::stride,:] time_steps, wait = int(N_t/stride), 80 math_animation(Data,time_steps,view,wait)
def example3(): L, T = 1., 2 N_x, N_t = 200, 440 m, u_0 = 20, 1/3. def f(x): return u_0*np.exp(-m**2.*(x-1./2.)**2.) def g(x): return np.zeros(x.shape) view = [-.1,L+.1,-.5,.5] Data = wave_1d(f,g,L,N_x,T,N_t,view) stride = 2 Data = Data[0], Data[1][::stride,:] time_steps, wait = int(N_t/stride), 80 math_animation(Data,time_steps,view,wait)
def Burgers():
def bump(x):
sigma, mu = 1., 0.
gaussian = (1. / (sigma * np.sqrt(2 * np.pi))) * np.exp(
(-1. / 2.) * ((x - mu) / sigma)**2.)
return gaussian * (3.5 * np.sin(3 * x) + 3.5)
def pde_matrix_func(Y2, Y1):
out = np.zeros(len(Y1))
c1 = (theta * delta_t) / (2. * delta_x)
c2 = (theta * delta_t) / (delta_x**2.)
c3 = ((1. - theta) * delta_t) / (2. * delta_x)
c4 = ((1. - theta) * delta_t) / (delta_x**2.)
out[1:-1] = (Y2[1:-1] + c1 * (Y2[1:-1] - s) * (Y2[2:] - Y2[:-2]) - c2 *
(Y2[2:] - 2. * Y2[1:-1] + Y2[:-2]) - Y1[1:-1] + c3 *
(Y1[1:-1] - s) * (Y1[2:] - Y1[:-2]) - c4 *
(Y1[2:] - 2. * Y1[1:-1] + Y1[:-2]))
out[0] = Y2[0] - Y1[0]
out[-1] = Y2[-1] - Y1[-1]
# for j in range(len(Y1)-1,len(Y1)): out[j] = Y2[j]-Y1[j]
return out
L, T = 20., 1.
u_m, u_p = 5., 1.
s, a = (u_m + u_p) / 2., (u_m - u_p) / 2.
n, time_steps = 150, 350
delta_x, delta_t = 2. * L / n, T / (n)
x = np.linspace(-L, L, n)
wave, perturb = s - a * np.tanh((a / 2.) * x), bump(x)
theta = 1 / 2.
U = np.zeros((time_steps, n))
U[0, :] = wave + perturb
U[0, 0], U[0, -1] = u_m, u_p
for j in xrange(1, time_steps):
U[j, :] = fsolve(lambda X, Y=U[j - 1, :]: pde_matrix_func(X, Y),
U[j - 1, :])
print j
view = [-L, L, u_p - 1, u_m + 1]
stride = 1
Data = x, U[::stride, :], wave
time_steps, wait = int(time_steps / stride), 30
math_animation(Data, time_steps, view, wait)
# plt.plot(x,wave,'-k',linewidth=2.,label=r'$\hat{u}$')
# plt.plot(x,wave+perturb,'-r',linewidth=2.,label=r'$v$')
# plt.axis([-20,20,0,6])
# # lg = legend.Legend()
# # lg.draw_frame(False)
# plt.legend(loc='best',fontsize=20)
# # plt.legend.draw_frame(False)
# plt.savefig('Perturbed Wave.pdf')
# plt.show()
return
def Burgers():
def bump(x):
sigma, mu = 1., 0.
gaussian = (1./(sigma*np.sqrt(2*np.pi) ) )*np.exp((-1./2.)*((x-mu)/sigma)**2.)
return gaussian*(3.5*np.sin(3*x)+ 3.5)
def pde_matrix_func(Y2,Y1):
out = np.zeros(len(Y1))
c1 = (theta*delta_t)/(2.*delta_x)
c2 = (theta*delta_t)/(delta_x**2.)
c3 = ((1.-theta)*delta_t)/(2.*delta_x)
c4 = ((1.-theta)*delta_t)/(delta_x**2.)
out[1:-1] = ( Y2[1:-1] + c1*(Y2[1:-1] - s)*(Y2[2:] - Y2[:-2]) -
c2*(Y2[2:] - 2.*Y2[1:-1] + Y2[:-2])-
Y1[1:-1] + c3*(Y1[1:-1] - s)*(Y1[2:] - Y1[:-2]) -
c4*(Y1[2:] - 2.*Y1[1:-1] + Y1[:-2])
)
out[0] = Y2[0]-Y1[0]
out[-1] = Y2[-1]-Y1[-1]
# for j in range(len(Y1)-1,len(Y1)): out[j] = Y2[j]-Y1[j]
return out
L, T = 20., 1.
u_m,u_p = 5.,1.
s, a = (u_m + u_p)/2., (u_m - u_p)/2.
n, time_steps = 150, 350
delta_x, delta_t = 2.*L/n, T/(n)
x = np.linspace(-L,L,n)
wave, perturb = s - a*np.tanh((a/2.)*x), bump(x)
theta = 1/2.
U = np.zeros( (time_steps, n) )
U[0,:] = wave+perturb
U[0,0], U[0,-1] = u_m, u_p
for j in xrange(1, time_steps):
U[j,:] = fsolve(lambda X,Y= U[j-1,:]: pde_matrix_func(X,Y), U[j-1,:] )
print j
view = [-L,L,u_p-1,u_m+1]
stride = 1
Data = x, U[::stride,:], wave
time_steps, wait = int(time_steps/stride), 30
math_animation(Data,time_steps,view,wait)
# plt.plot(x,wave,'-k',linewidth=2.,label=r'$\hat{u}$')
# plt.plot(x,wave+perturb,'-r',linewidth=2.,label=r'$v$')
# plt.axis([-20,20,0,6])
# # lg = legend.Legend()
# # lg.draw_frame(False)
# plt.legend(loc='best',fontsize=20)
# # plt.legend.draw_frame(False)
# plt.savefig('Perturbed Wave.pdf')
# plt.show()
return
