def kdv(u, t, L):
"""Differential equations for the KdV equation, discretized in x."""
# Compute the x derivatives using the pseudo-spectral method.
ux = psdiff(u, period=L)
uxxx = psdiff(u, period=L, order=3)
# Compute du/dt.
dudt = -6*u*ux - uxxx
return dudt
def kdv(u, t, L):
"""Differential equations for the KdV equation, discretized in x."""
# Compute the x derivatives using the pseudo-spectral method.
ux = psdiff(u, period=L)
uxxx = psdiff(u, period=L, order=3)
# Compute du/dt.
dudt = -6 * u * ux - uxxx
return dudt
def L_dU_perp(dU_perp): # Function used with fsolve to calculate du_perp_dx at x = x_min du_perp_r = dU_perp[0:nz] du_perp_i = dU_perp[nz:] du_perp = du_perp_r + 1j * du_perp_i L1 = omega ** 2 * rhox(x_min,z) du_perp_zz = psdiff(du_perp, order=2, period=2*Lz) L_du_perp = np.cos(alpha) ** 2 * du_perp_zz + omega ** 2 / vA(x_min,z) ** 2 * du_perp nabla_perp_db_par = 1j * k_perp * db_par_x_min - np.sin(alpha) * psdiff(db_par_x_min, period=2*Lz) res = L_du_perp + L1 * u_perp_x_min - 1j * omega * nabla_perp_db_par return np.concatenate((np.real(res), np.imag(res)))
def residual(x,U,dU): ux_r = U[0:ct.nz] ux_i = U[ct.nz:2*ct.nz] b_par_r = U[2*ct.nz:3*ct.nz] b_par_i = U[3*ct.nz:4*ct.nz] u_perp_r = U[4*ct.nz:5*ct.nz] u_perp_i = U[5*ct.nz:] dux_r = dU[0:ct.nz] dux_i = dU[ct.nz:2*ct.nz] db_par_r = dU[2*ct.nz:3*ct.nz] db_par_i = dU[3*ct.nz:4*ct.nz] du_perp_r = dU[4*ct.nz:5*ct.nz] du_perp_i = dU[5*ct.nz:] ux = ux_r + 1j * ux_i b_par = b_par_r + 1j * b_par_i u_perp = u_perp_r + 1j * u_perp_i dux = dux_r + 1j * dux_i db_par = db_par_r + 1j * db_par_i du_perp = du_perp_r + 1j * du_perp_i nabla_perp_b_par = 1j * ct.k_perp * b_par - np.sin(ct.alpha) * psdiff(b_par, period = 2 * ct.Lz) nabla_perp_u_perp = 1j * ct.k_perp * u_perp - np.sin(ct.alpha) * psdiff(u_perp, period = 2 * ct.Lz) L_ux = np.cos(ct.alpha) ** 2 * psdiff(ux, order = 2, period = 2 * ct.Lz) \ + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * ux L_u_perp = np.cos(ct.alpha) ** 2 * psdiff(u_perp, order = 2, period = 2 * ct.Lz) \ + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp # nabla_perp_b_par = 1j * ct.k_perp * b_par - 1j * np.sin(ct.alpha) * b_par # nabla_perp_u_perp = 1j * ct.k_perp * u_perp - 1j * np.sin(ct.alpha) * u_perp # L_ux = -ct.kz ** 2 * np.cos(ct.alpha) ** 2 * ux \ # + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * ux # L_u_perp = -ct.kz ** 2 * np.cos(ct.alpha) ** 2 * u_perp \ # + ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp res = np.zeros(6 * ct.nz) res[0:ct.nz] = np.real(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[ct.nz:2*ct.nz] = np.imag(dux + 1j * ct.omega * b_par + nabla_perp_u_perp) res[2*ct.nz:3*ct.nz] = np.real(db_par + 1j / ct.omega * L_ux) res[3*ct.nz:4*ct.nz] = np.imag(db_par + 1j / ct.omega * L_ux) res[4*ct.nz:5*ct.nz] = np.real(L_u_perp - 1j * ct.omega * nabla_perp_b_par) res[5*ct.nz:] = np.imag(L_u_perp - 1j * ct.omega * nabla_perp_b_par) return res
def residual(x, U, dU):
ux_r = U[0:nz]
ux_i = U[nz:2 * nz]
b_par_r = U[2 * nz:3 * nz]
b_par_i = U[3 * nz:4 * nz]
u_perp_r = U[4 * nz:5 * nz]
u_perp_i = U[5 * nz:]
dux_r = dU[0:nz]
dux_i = dU[nz:2 * nz]
db_par_r = dU[2 * nz:3 * nz]
db_par_i = dU[3 * nz:4 * nz]
du_perp_r = dU[4 * nz:5 * nz]
du_perp_i = dU[5 * nz:]
ux = ux_r + 1j * ux_i
b_par = b_par_r + 1j * b_par_i
u_perp = u_perp_r + 1j * u_perp_i
dux = dux_r + 1j * dux_i
db_par = db_par_r + 1j * db_par_i
du_perp = du_perp_r + 1j * du_perp_i
nabla_perp_b_par = 1j * k_perp * b_par - np.sin(alpha) * psdiff(
b_par, period=2 * Lz)
nabla_perp_u_perp = 1j * k_perp * u_perp - np.sin(alpha) * psdiff(
u_perp, period=2 * Lz)
L_ux = np.cos(alpha) ** 2 * psdiff(ux, order = 2, period = 2 * Lz) \
+ omega ** 2 / vA(x,z) ** 2 * ux
L_u_perp = np.cos(alpha) ** 2 * psdiff(u_perp, order = 2, period = 2 * Lz) \
+ omega ** 2 / vA(x,z) ** 2 * u_perp
res = np.zeros(6 * nz)
res[0:nz] = np.real(dux + 1j * omega * b_par + nabla_perp_u_perp)
res[nz:2 * nz] = np.imag(dux + 1j * omega * b_par + nabla_perp_u_perp)
res[2 * nz:3 * nz] = np.real(db_par + 1j / omega * L_ux)
res[3 * nz:4 * nz] = np.imag(db_par + 1j / omega * L_ux)
res[4 * nz:5 * nz] = np.real(L_u_perp - 1j * omega * nabla_perp_b_par)
res[5 * nz:] = np.imag(L_u_perp - 1j * omega * nabla_perp_b_par)
return res
def dUdt(U, t):
"""
Compute dUdt, given U.
Solve the algebraic system of equations to find the variables.
(Given U, compute F and S
:param U: Vectors of u
:return: F, S
"""
## Compute F
#F = np.zeros([5, x.size])
F = np.zeros_like(U)
F[0] = U[1] # 'rho * u'
# Compute the velocity at every element
u = U[1] / U[0] # u = rho * u / rho
#print(u)
# Compute the energy (temperature) at every element # Validated
E = U[2] / U[
0] - u**2 / 2.0 # 'e = rho * ((e + u**2/2.0))/rho - u**2/2.0'
#print(f'E = {E}')
# e(p, v, lambd, phi)
#F[1] = U[0] * u**2 + p
# Compute the porosity at every element
PHI = U[3] / U[0] # phi = (rho * phi) / rho
#print(f'Phi = {Phi}')
# Compute the reaction progress at every element
LAMBD = U[4] / U[0] # lambd = (rho * lambd) / rho
#print(f'Lambd = {Lambd}')
F[3] = F[0] * PHI # 'rho * u * phi' # also U[1] * U[3]/U[0]
F[4] = F[0] * LAMBD # 'rho * u * Lambd' # also U[1] * U[4]/U[0]
# Compute the pressure
# Compute the specific volume from the density
V = U[0]**(-1) # specific volume
#TODO: Try to vectorize
## P = p_from_e(E, V, LAMBD, PHI)
P = np.zeros(np.shape(U)[1])
for ind in range(np.shape(U)[1]): # TODO: Extremely slow
P[ind] = p_from_e(E[ind], V[ind], LAMBD[ind], PHI[ind])
#print(f'P = {P}')
F[1] = U[0] * u**2 + P # rho * u^2 + p
#print(f'F[1] = {F[1]}')
F[2] = F[0] * (E + u**2 / 2.0 + P / U[0])
# 'rho * u * (e + u^2/2 + p/rho)
## Compute S
#S = np.zeros([5, x.size])
S = np.zeros_like(U)
R_phi = r_phi(P, PHI)
#print(f'R_phi = {R_phi}')
#print(f'r_phi(P[0], Phi[0]) = {r_phi(P[0], Phi[0])}')
#print(f'r_phi({P[0]}, {Phi[0]}) = {r_phi(P[0], Phi[0])}')
R_lambd = r_lambda(P, LAMBD)
#print(f'R_lambd = {R_lambd}')
S[3] = U[0] * R_phi # rho * r_phi
S[4] = U[0] * R_lambd # rho * r_lam
#print(S)
# Gradient of F
dFdx0 = psdiff(F[0], period=L)
dFdx1 = psdiff(F[1], period=L)
dFdx2 = psdiff(F[2], period=L)
dFdx3 = psdiff(F[3], period=L)
dFdx4 = psdiff(F[4], period=L)
#print(dFdx4)
dFdx = np.array([
dFdx0,
dFdx1,
dFdx2,
dFdx3,
dFdx4,
])
d2Fdx0 = psdiff(F[0], order=2, period=L)
d2Fdx1 = psdiff(F[1], order=2, period=L)
d2Fdx2 = psdiff(F[2], order=2, period=L)
d2Fdx3 = psdiff(F[3], order=2, period=L)
d2Fdx4 = psdiff(F[4], order=2, period=L)
dF2dx = np.array([
d2Fdx0,
d2Fdx1,
d2Fdx2,
d2Fdx3,
d2Fdx4,
])
#print(dFdx)
#print(S - dFdx)
return S - dFdx - 0.1e-3 * dF2dx
nabla_perp0 = 1j * k_perp
beta0 = xi
# Calculate b_par0
L1 = omega**2 * rhox(0, z)
kzn = 2 * np.pi * fftfreq(nz, dz)
kzn[0] = 1 # To avoid division by zero, note that fft(f)[0] is approximately equal to zero anyway
f = beta0 * L1 / (1j * omega) * sol.sol(z)[0]
b_par0 = ifft(fft(f) / 1j / (k_perp - kzn * np.sin(alpha)))
ux_x_min = ux_ana(x_min, z)
b_par_x_min = b_par_ana(x_min, z)
u_perp_x_min = u_perp_ana(x_min, z)
nabla_perp_u_perp = 1j * k_perp * u_perp_x_min - np.sin(alpha) * psdiff(
u_perp_x_min, period=2 * Lz)
ux_zz = psdiff(ux_x_min, order=2, period=2 * Lz)
L_ux = np.cos(alpha)**2 * ux_zz + omega**2 / vA(x_min, z)**2 * ux_x_min
dux_x_min = -(1j * omega * b_par_x_min + nabla_perp_u_perp)
db_par_x_min = -1j / omega * L_ux
dU_perp_x_min = fsolve(
L_dU_perp,
np.concatenate((du_perp_ana(x_min, z).real, du_perp_ana(x_min, z).imag)))
U_x_min = np.concatenate((ux_x_min.real, \
ux_x_min.imag, \
b_par_x_min.real, \
b_par_x_min.imag, \
u_perp_x_min.real, \
u_perp_x_min.imag))