def func0(): nu, L = .1, 1. A, B, T = -L, L, 2. time_steps = 200 delta_t = T/time_steps u_m,u_p = 5.,1. s, a = (u_m + u_p)/2., (u_m - u_p)/2. N = 200 D,x = cheb(N) M = D.dot(D) M[0,:], M[-1,:] = 0., 0. M[0,0], M[-1,-1] = 1., 1. D2 = D[1:-1,:] bcs = np.zeros(N+1) bcs[0], bcs[-1] = 1., 5. def cap_F(u): out = M.dot(u) - bcs out[1:-1] += (L/nu)*(s-u[1:-1])*D2.dot(u) return out guess = s - a*np.tanh(.6*(a/(2.*nu))*x) sol = root(cap_F, guess ) if sol.success: soln = s - a*np.tanh((a/(2.*nu))*(1./2)*(B+A + (B-A)*x)) plt.plot((1./2)*(B+A + (B-A)*x),soln,'-k',linewidth=2.0,label='True solution') plt.plot((1./2)*(B+A + (B-A)*x),sol.x,'-r',linewidth=2.0,label='Numerical solution') plt.plot(x,guess,'*r',linewidth=2.0,label='Guess') plt.axis([-L,L,.5,5.5]) plt.legend(loc='best') plt.show() return
def func0(): nu, L = .1, 1. A, B, T = -L, L, 2. time_steps = 200 delta_t = T/time_steps u_m,u_p = 5.,1. s, a = (u_m + u_p)/2., (u_m - u_p)/2. N = 200 D,x = cheb(N) M = D.dot(D) M[0,:], M[-1,:] = 0., 0. M[0,0], M[-1,-1] = 1., 1. D2 = D[1:-1,:] bcs = np.zeros(N+1) bcs[0], bcs[-1] = 1., 5. def cap_F(u): out = M.dot(u) - bcs out[1:-1] += (L/nu)*(s-u[1:-1])*D2.dot(u) return out guess = s - a*np.tanh(.6*(a/(2.*nu))*x) sol = root(cap_F, guess ) if sol.success: soln = s - a*np.tanh((a/(2.*nu))*(1./2)*(B+A + (B-A)*x)) plt.plot((1./2)*(B+A + (B-A)*x),soln,'-k',linewidth=2.0,label='True solution') plt.plot((1./2)*(B+A + (B-A)*x),sol.x,'-r',linewidth=2.0,label='Numerical solution') plt.plot(x,guess,'*r',linewidth=2.0,label='Guess') plt.axis([-L,L,.5,5.5]) plt.legend(loc='best') plt.show() return
def exercise2(): # Solves the Poisson boundary value problem u''(x) = f(x), u(-1) = u(1) = 0 def f(x): return np.exp(2.*x) #np.exp(4.*x) def anal_sol(x): # return (np.exp(4.*x) - np.sinh(4.)*x - np.cosh(4.))/16. return (np.exp(2.*x) - np.sinh(2.)*x - np.cosh(2.))/4. N = 3 D,x = cheb(N) A = np.dot(D,D)[1:N:1,1:N:1] u = np.zeros(x.shape); u[1:N] = solve(A,f(x[1:N:1])) # print "\nDx = \n", x[1:N:1] # print "\nA = \n", A xx = np.linspace(-1,1,50) uu = (np.poly1d(np.polyfit(x,u,N)))(xx) plt.plot(x,u,'*r'); plt.plot(xx,uu,'-r') plt.plot(xx,anal_sol(xx),'-k') plt.axis([-1.,1.,-2.5,.5]); plt.show() print "Max error is ", np.max(np.abs(uu - anal_sol(xx))) return
def Burgers(): nu, L = .1, 2. u_m,u_p = 5.,1. s, a = (u_m + u_p)/2., (u_m - u_p)/2. def guess(x): return s - a*np.tanh(a/(2.*nu)*x) N = 50 D,x = cheb(N) M = D.dot(D) M[0,:], M[-1,:] = 0., 0. M[0,0], M[-1,-1] = 1., 1. D2 = D[1:-1,:] u_bcs = np.zeros(N+1) y_bcs = np.zeros(N+1) u_bcs[0], u_bcs[-1] = u_p, s y_bcs[0], y_bcs[-1] = u_m, s def cap_F(u): out = np.zeros(2*(N+1)) out1, out2 = out[:N+1], out[N+1:] u1, u2 = u[:N+1], u[N+1:] out1 += M.dot(u1) - u_bcs out1[1:-1] += (.5*L/nu)*(s - u1[1:-1])*D2.dot(u1) out2 += M.dot(u2) - y_bcs out2[1:-1] -= (.5*L/nu)*(s - u2[1:-1])*D2.dot(u2) return out array = np.empty(2*(N+1)) array[:N+1] = guess(.4*L/2.*(x+1.)) array[N+1:] = guess(-(.3)*L/2.*(x+1.)) sol = root(cap_F, array ) if sol.success: plt.plot(x,sol.x[:N+1],'-*k',linewidth=2.) plt.plot(x,sol.x[N+1:],'-*k',linewidth=2.) # plt.plot(x,array[:N+1],'-*r') # plt.plot(x,array[N+1:],'-*r') plt.axis([-1,1,u_p-.5, u_m+.5]) plt.show() x_orig = L/2.*(x+1.) plt.plot(x_orig,sol.x[:N+1],'-ok',markersize=3.,linewidth=2.,label='Numerical solution') plt.plot(-x_orig,sol.x[N+1:],'-ok',markersize=3.,linewidth=2.) plt.plot(x_orig,array[:N+1],'-or',markersize=3.,linewidth=2.,label='Guess') plt.plot(-x_orig,array[N+1:],'-or',markersize=3.,linewidth=2.) plt.axis([-L,L,u_p-.5, u_m+.5]) plt.legend(loc='best') plt.show() return
def Burgers(): nu, L = .1, 2. u_m,u_p = 5.,1. s, a = (u_m + u_p)/2., (u_m - u_p)/2. def guess(x): return s - a*np.tanh(a/(2.*nu)*x) N = 50 D,x = cheb(N) M = D.dot(D) M[0,:], M[-1,:] = 0., 0. M[0,0], M[-1,-1] = 1., 1. D2 = D[1:-1,:] u_bcs = np.zeros(N+1) y_bcs = np.zeros(N+1) u_bcs[0], u_bcs[-1] = u_p, s y_bcs[0], y_bcs[-1] = u_m, s def cap_F(u): out = np.zeros(2*(N+1)) out1, out2 = out[:N+1], out[N+1:] u1, u2 = u[:N+1], u[N+1:] out1 += M.dot(u1) - u_bcs out1[1:-1] += (.5*L/nu)*(s - u1[1:-1])*D2.dot(u1) out2 += M.dot(u2) - y_bcs out2[1:-1] -= (.5*L/nu)*(s - u2[1:-1])*D2.dot(u2) return out array = np.empty(2*(N+1)) array[:N+1] = guess(.4*L/2.*(x+1.)) array[N+1:] = guess(-(.3)*L/2.*(x+1.)) sol = root(cap_F, array ) if sol.success: plt.plot(x,sol.x[:N+1],'-*k',linewidth=2.) plt.plot(x,sol.x[N+1:],'-*k',linewidth=2.) # plt.plot(x,array[:N+1],'-*r') # plt.plot(x,array[N+1:],'-*r') plt.axis([-1,1,u_p-.5, u_m+.5]) plt.show() x_orig = L/2.*(x+1.) plt.plot(x_orig,sol.x[:N+1],'-ok',markersize=3.,linewidth=2.,label='Numerical solution') plt.plot(-x_orig,sol.x[N+1:],'-ok',markersize=3.,linewidth=2.) plt.plot(x_orig,array[:N+1],'-or',markersize=3.,linewidth=2.,label='Guess') plt.plot(-x_orig,array[N+1:],'-or',markersize=3.,linewidth=2.) plt.axis([-L,L,u_p-.5, u_m+.5]) plt.legend(loc='best') plt.show() return
def nonzeroDirichlet():
# Solves the Poisson boundary value problem
# u''(x) + u'(x) = f(x),
# u(-1) = a
# u(1) = b
a, b = 2, -1
def f(x):
return -(b - a) / 2. * np.ones(x.shape) + np.exp(3. * x)
def G(x):
return (b - a) / 2. * np.ones(x.shape) * x + (b + a) / 2. * np.ones(
x.shape)
def anal_sol(x):
# out = ( np.exp(-x)/( 12.*(-1 +np.exp(1))*(1 + np.exp(1))*np.exp(3)) *
# (-12.*a*np.exp(x+3.)+12.*a*np.exp(4) + 12.*b*np.exp(x+5)-12*b*np.exp(4) -
# np.exp(4*x+3) + np.exp(4*x+5) - np.exp(x+8 )+ np.exp(x)- np.exp(1) + np.exp(7)
# )
# )
N, B = np.array([[1., np.exp(1)],
[1, np.exp(-1)]]), np.array([[a - np.exp(-3) / 12.],
[b - np.exp(3) / 12.]])
C = solve(N, B)
return C[0] + C[1] * np.exp(-x) + np.exp(3. * x) / 12.
N = 7
D, x = cheb(N)
A = np.dot(D, D)[1:N:1, 1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A + D[1:N:1, 1:N:1], f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x, u, N)))(xx)
print "Max error is ", np.max(np.abs(uu + G(xx) - anal_sol(xx)))
# plt.plot(x,u+G(x), '*k')
# plt.plot(xx,uu+G(xx), '-r')
plt.plot(xx, anal_sol(xx))
plt.xlabel('$x$')
plt.ylabel('$u$')
# plt.axis([-1.,1.,-2.5,.5])
plt.savefig('nonzeroDirichlet.pdf')
plt.clf()
def deriv_matrix_exercise1(): def f(x): return np.exp(x)*np.cos(6.*x) def fp(x): return np.exp(x)*(np.cos(6.*x) - 6.*np.sin(6.*x)) N = 10 D,x = cheb(N) f_exact, fp_exact = f(x), fp(x) xx = np.linspace(-1,1,200) plt.plot(xx,fp(xx),'-k') # Exact Derivative uu = (np.poly1d(np.polyfit(x, D.dot(f(x) ), N)))(xx) plt.plot(x,D.dot(f(x)),'*r') # Approximation to derivative at grid points plt.plot(xx,uu,'-r') # Approximation to derivative at other values plt.show() print "max error = \n", np.max(np.abs( D.dot(f(x)) - fp(x) )) return
def nonzeroDirichlet():
# Solves the Poisson boundary value problem
# u''(x) + u'(x) = f(x),
# u(-1) = a
# u(1) = b
a, b = 2, -1
def f(x):
return -(b-a)/2.*np.ones(x.shape)+ np.exp(3.*x)
def G(x):
return (b-a)/2.*np.ones(x.shape) *x + (b+a)/2.*np.ones(x.shape)
def anal_sol(x):
# out = ( np.exp(-x)/( 12.*(-1 +np.exp(1))*(1 + np.exp(1))*np.exp(3)) *
# (-12.*a*np.exp(x+3.)+12.*a*np.exp(4) + 12.*b*np.exp(x+5)-12*b*np.exp(4) -
# np.exp(4*x+3) + np.exp(4*x+5) - np.exp(x+8 )+ np.exp(x)- np.exp(1) + np.exp(7)
# )
# )
N, B = np.array([[1.,np.exp(1)],[1,np.exp(-1)]]), np.array([[a-np.exp(-3)/12.], [b-np.exp(3)/12.]])
C = solve(N,B)
return C[0] + C[1]*np.exp(-x) + np.exp(3.*x)/12.
N = 7
D, x = cheb(N)
A = np.dot(D, D)[1:N:1,1:N:1]
u = np.zeros(x.shape)
u[1:N] = solve(A + D[1:N:1,1:N:1], f(x[1:N:1]))
xx = np.linspace(-1, 1, 50)
uu = (np.poly1d(np.polyfit(x,u,N)))(xx)
print "Max error is ", np.max(np.abs(uu + G(xx) - anal_sol(xx)))
# plt.plot(x,u+G(x), '*k')
# plt.plot(xx,uu+G(xx), '-r')
plt.plot(xx, anal_sol(xx))
plt.xlabel('$x$')
plt.ylabel('$u$')
# plt.axis([-1.,1.,-2.5,.5])
plt.savefig('nonzeroDirichlet.pdf')
plt.clf()
def exercise3(): # Solves the Poisson boundary value problem # u''(x) + u'(x) = f(x), # u(-1) = a # u(1) = b a, b = 2, -1 def f(x): return -(b-a)/2.*np.ones(x.shape)+ np.exp(3.*x) def G(x): return (b-a)/2.*np.ones(x.shape) *x + (b+a)/2.*np.ones(x.shape) def anal_sol(x): out = ( np.exp(-x)/( 12.*(-1 +np.exp(1))*(1 + np.exp(1))*np.exp(3) ) * ( -12.*a*np.exp(x+3.)+12.*a*np.exp(4) + 12.*b*np.exp(x+5)-12*b*np.exp(4) - np.exp(4*x+3) + np.exp(4*x+5) - np.exp(x+8 )+ np.exp(x)- np.exp(1) + np.exp(7) ) ) return out N = 8 D,x = cheb(N) A = np.dot(D,D)[1:N:1,1:N:1] u = np.zeros(x.shape); u[1:N] = solve(A + D[1:N:1,1:N:1],f(x[1:N:1])) # print "\nDx = \n", x[1:N:1] # print "\nA = \n", A xx = np.linspace(-1,1,50) uu = (np.poly1d(np.polyfit(x,u,N)))(xx) plt.plot(x,u+G(x),'*k') plt.plot(xx,uu+G(xx),'-r') plt.plot(xx,anal_sol(xx),'-k') # plt.axis([-1.,1.,-2.5,.5]); plt.show() print "Max error is ", np.max(np.abs(uu+G(xx) - anal_sol(xx))) return
def Solitons(): # nu, L = .1, 2. # u_m,u_p = 5.,1. # s, a = (u_m + u_p)/2., (u_m - u_p)/2. L = 120. s = .5 def guess(x): return (3.*s/2)*np.cosh( (math.sqrt(s/2.)/2.)*(x+3.5) )**(-2.) # x = np.linspace(-1,1,400) # plt.plot(x,guess(x),'-k') # plt.show() # import sys; sys.exit() # N = 50 # D,x = cheb(N) # M = D.dot(D) # M[0,:], M[-1,:] = 0., 0. # M[0,0], M[-1,-1] = 1., 1. # D2 = D[1:-1,:] N = 50 D1, x = cheb(N) D2 = D1.dot(D1) D3 = D2.dot(D1) D2[0,:], D2[-1,:] = 0., 0. D2[0,0], D2[-1,-1] = 1., 1. D1mod = D1[1:-1,:] u_bcs = np.zeros(N+1) y_bcs = np.zeros(N+1) u_bcs[0], u_bcs[-1] = 0., (3./3)*s y_bcs[0], y_bcs[-1] = 0., (3./3)*s def cap_F(u): out = np.zeros(2*(N+1)) out1, out2 = out[:N+1], out[N+1:] u1, u2 = u[:N+1], u[N+1:] out1 += D2.dot(u1) - u_bcs out1[1:-1] -= (s - u1[1:-1]/2.)*u1[1:-1] out2 += D2.dot(u2) - y_bcs out2[1:-1] -= (s - u2[1:-1]/2.)*u2[1:-1] return out array = np.empty(2*(N+1)) array[:N+1] = guess(.4*L/2.*(x+1.)) array[N+1:] = guess(-(.3)*L/2.*(x+1.)) print np.min(array[N+1:]), np.min(array[:N+1]) sol = root(cap_F, array ) if sol.success: # print sol plt.plot(x,sol.x[:N+1],'-*k',linewidth=2.) plt.plot(x,sol.x[N+1:],'-*k',linewidth=2.) # plt.plot(x,array[:N+1],'-*r') # plt.plot(x,array[N+1:],'-*r') # plt.axis([-1,1,0.-.01, 2.*s]) plt.show() x_orig = L/2.*(x+1.) plt.plot(x_orig,sol.x[:N+1],'-ok',markersize=3.,linewidth=2.,label='Numerical solution') plt.plot(-x_orig,sol.x[N+1:],'-ok',markersize=3.,linewidth=2.) plt.plot(x_orig,array[:N+1],'-or',markersize=3.,linewidth=2.,label='Guess') plt.plot(-x_orig,array[N+1:],'-or',markersize=3.,linewidth=2.) # plt.axis([-L,L,0.-.01, 2.*s]) plt.legend(loc='best') plt.show() return
def Solitons():
# nu, L = .1, 2.
# u_m,u_p = 5.,1.
# s, a = (u_m + u_p)/2., (u_m - u_p)/2.
L = 120.
s = .5
def guess(x):
return (3. * s / 2) * np.cosh(
(math.sqrt(s / 2.) / 2.) * (x + 3.5))**(-2.)
# x = np.linspace(-1,1,400)
# plt.plot(x,guess(x),'-k')
# plt.show()
# import sys; sys.exit()
# N = 50
# D,x = cheb(N)
# M = D.dot(D)
# M[0,:], M[-1,:] = 0., 0.
# M[0,0], M[-1,-1] = 1., 1.
# D2 = D[1:-1,:]
N = 50
D1, x = cheb(N)
D2 = D1.dot(D1)
D3 = D2.dot(D1)
D2[0, :], D2[-1, :] = 0., 0.
D2[0, 0], D2[-1, -1] = 1., 1.
D1mod = D1[1:-1, :]
u_bcs = np.zeros(N + 1)
y_bcs = np.zeros(N + 1)
u_bcs[0], u_bcs[-1] = 0., (3. / 3) * s
y_bcs[0], y_bcs[-1] = 0., (3. / 3) * s
def cap_F(u):
out = np.zeros(2 * (N + 1))
out1, out2 = out[:N + 1], out[N + 1:]
u1, u2 = u[:N + 1], u[N + 1:]
out1 += D2.dot(u1) - u_bcs
out1[1:-1] -= (s - u1[1:-1] / 2.) * u1[1:-1]
out2 += D2.dot(u2) - y_bcs
out2[1:-1] -= (s - u2[1:-1] / 2.) * u2[1:-1]
return out
array = np.empty(2 * (N + 1))
array[:N + 1] = guess(.4 * L / 2. * (x + 1.))
array[N + 1:] = guess(-(.3) * L / 2. * (x + 1.))
print np.min(array[N + 1:]), np.min(array[:N + 1])
sol = root(cap_F, array)
if sol.success:
# print sol
plt.plot(x, sol.x[:N + 1], '-*k', linewidth=2.)
plt.plot(x, sol.x[N + 1:], '-*k', linewidth=2.)
# plt.plot(x,array[:N+1],'-*r')
# plt.plot(x,array[N+1:],'-*r')
# plt.axis([-1,1,0.-.01, 2.*s])
plt.show()
x_orig = L / 2. * (x + 1.)
plt.plot(x_orig,
sol.x[:N + 1],
'-ok',
markersize=3.,
linewidth=2.,
label='Numerical solution')
plt.plot(-x_orig, sol.x[N + 1:], '-ok', markersize=3., linewidth=2.)
plt.plot(x_orig,
array[:N + 1],
'-or',
markersize=3.,
linewidth=2.,
label='Guess')
plt.plot(-x_orig, array[N + 1:], '-or', markersize=3., linewidth=2.)
# plt.axis([-L,L,0.-.01, 2.*s])
plt.legend(loc='best')
plt.show()
return
