plt.xlabel(r'Distance $(m)$')
plt.ylabel(r'Concentration $ \frac{kg}{m} $')
plt.pause(3)
plt.close(1)
# =============================================================================
# Solving first timestep with forward Euler
# =============================================================================
# Generating analytical solution
Ca = AN.difuana(M, L, Dx, xn, xo, T0 + dT)
# Estimating solution C1 in t
CL1 = Ca[0]
CR1 = Ca[len(xn) - 1]
spa = AUX.FVev_sp(C, Dx, dx, CL1, CR1)
C1 = C + dT * spa
# =============================================================================
# Starting time loop
# =============================================================================
plt.ion()
plt.figure(1, figsize=(11, 8.5))
style.use('ggplot')
for t in range(2, npt + 1):
# Generating analytical solution
Ca = AN.difuana(M, L, Dx, xn, xo, T0 + t * dT)
# Starting time loop
# =============================================================================
plt.ion()
plt.figure(1, figsize=(11, 8.5))
style.use('ggplot')
for t in range(1, npt + 1):
# Generating analytical solution
Ca = AN.difuana(M, L, Dx, xn, xo, T0 + t * dT)
# Estimating solution C1 in t
CL = AN.difuana(M, L, Dx, X0, xo, T0 + (t - 1) * dT)
CR = AN.difuana(M, L, Dx, XL, xo, T0 + (t - 1) * dT)
spa = AUX.FVev_sp(C, Dx, dx, CL, CR)
C1 = C + dT * spa
# Implicit part of the solution
# Imposing boundary conditions
C[0] = C[0] + AN.difuana(M, L, Dx, X0, xo, T0 + t * dT) * Sx * 2
C[len(xn) - 1] = C[len(xn) - 1] + AN.difuana(M, L, Dx, XL, xo, T0 + t * dT)\
* Sx * 2
# Solving linear system
C1i = spsolve(K, C)
# Crank-Nicholson ponderation
C1t = (1 - theta) * C1 + theta * C1i
def FVM1D(Sx, N):
# ==============================================================================
# Variable declaration
# ==============================================================================
X0 = 0. # Initial x coordinate
XL = 5. # Final x coordinate
M = 10 # Mass injected
xo = 2 # Injection coordinate
Dx = 0.3 # Diffusion coefficient
T0 = 1. # Initial time
Tf = 5. # Final time
# ==============================================================================
# Numerical parameters
# theta = 0 --> Fully explicit
# theta = 1 --> Fully implicit
# ==============================================================================
# Sx = 0.3
theta = 0.5 # C-N ponderation factor
# N = 41 # Volumes in the domain
L = XL - X0 # Domain length
dx = L / N # Calculating spacing
xn = np.zeros(N) # Node coordinates vector
# Filling vector of node coordinates
for ic in range(0, N):
xn[ic] = ic * dx + dx / 2
# Calculating dT and number of timesteps
dT = Sx * (dx**2) / (Dx) # Calculating timestep size
npt = int(np.ceil((Tf - T0) / (dT))) # Number of timesteps
# Generating vector that stores error in time
ert = np.zeros(int(npt) + 1)
# ==============================================================================
# Generating matrix for implicit solver
# ==============================================================================
K = SP.lil_matrix((N, N))
K[0, 0] = 1 + 3 * Sx
K[0, 1] = -Sx
K[N - 1, N - 1] = 1 + 3 * Sx
K[N - 1, N - 2] = -Sx
for i in range(1, N - 1):
K[i, i] = 1 + 2 * Sx
K[i, i + 1] = -Sx
K[i, i - 1] = -Sx
# ==============================================================================
# Generating initial condition
# ==============================================================================
C = AN.difuana(M, L, Dx, xn, xo, T0)
C1 = np.zeros(N)
Cmax = np.max(C)
# # Plotting initial condition
# plt.ion()
# plt.figure(1, figsize=(11, 8.5))
# style.use('ggplot')
#
# plt.subplot(1, 1, 1)
# plt.plot(xn, C)
# plt.title('Initial condition')
# plt.xlabel(r'Distance $(m)$')
# plt.ylabel(r'Concentration $ \frac{kg}{m} $')
# =============================================================================
# Starting time loop
# =============================================================================
# plt.ion()
# plt.figure(1, figsize=(11, 8.5))
# style.use('ggplot')
for t in range(1, npt + 1):
# Generating analytical solution
Ca = AN.difuana(M, L, Dx, xn, xo, T0 + t * dT)
# Estimating solution C1 in t
CL = AN.difuana(M, L, Dx, X0, xo, T0 + (t - 1) * dT)
CR = AN.difuana(M, L, Dx, XL, xo, T0 + (t - 1) * dT)
spa = AUX.FVev_sp(C, Dx, dx, CL, CR)
C1 = C + dT * spa
# Implicit part of the solution
# Imposing boundary conditions
C[0] = C[0] + AN.difuana(M, L, Dx, X0, xo, T0 + t * dT) * Sx * 2
C[len(xn) - 1] = C[len(xn) - 1] + AN.difuana(M, L, Dx, XL, xo, T0 + t * dT)\
* Sx * 2
# Solving linear system
C1i = spsolve(K, C)
# Crank-Nicholson ponderation
C1t = (1 - theta) * C1 + theta * C1i
# Estimating error
err = np.abs(C1t - Ca)
ert[t] = np.linalg.norm(err)
# # Plotting numerical solution and comparison with analytical
# plt.clf()
#
# plt.subplot(2, 2, 1)
# plt.plot(xn, C1t, 'b')
# plt.xlim([X0, XL])
# plt.ylim([0, Cmax])
# plt.ylabel(r'Concentration $ \frac{kg}{m} $')
# plt.title('Numerical solution')
#
# plt.subplot(2, 2, 2)
# plt.plot(xn, Ca)
# plt.xlim([X0, XL])
# plt.ylim([0, Cmax])
# plt.title('Analytical solution')
#
# plt.subplot(2, 2, 3)
# plt.semilogy(xn, err)
# plt.xlim([X0, XL])
# plt.ylim([1e-8, 1e2])
# plt.ylabel('Absolute error')
# plt.title('Error')
#
# plt.subplot(2, 2, 4)
# plt.semilogy(np.linspace(T0, Tf, npt), ert)
# plt.xlim([T0 - 0.2, Tf + 0.2])
# plt.ylim([1e-8, 1e2])
# plt.title('Error evolution')
#
# plt.draw()
# plt.pause(0.2)
# Preparing for next timestep
C = C1t
return ert
