75 lines
2.2 KiB
Python
75 lines
2.2 KiB
Python
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# Task 1.4
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import numpy as np
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import scipy, scipy.sparse, scipy.linalg
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from matplotlib import pyplot as plt
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# plotting helper function
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def plot(C, x, t, name):
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plt.clf()
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plt.plot(x, C)
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plt.xlim([0, x[-1]])
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plt.ylim([0, 1.2])
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plt.title(f"{name} - time: {t: >12.0f}")
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plt.savefig(f"plots/{name}_{plot.fignr:0>5}.png")
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plot.fignr += 1
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# set static variable as function attribute
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plot.fignr = 1
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# and initialize a figure
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plt.figure(figsize = (8, 6), dpi = 100)
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# Config
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D = 1e-6 # diffusion coefficient
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h = 1 # space domain (max x size)
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T = 1e5 # solution end time
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nx = 50 # nr of space discretization points
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nt = 2000 # nr of time points
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# derived constants
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dx = h / (nx - 1) # space step size
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dt = T / (nt - 1) # time step size
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d = dt * D / dx**2 # stability/stepsize coefficient
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# setup matrices for implicit update scheme `A C^n+1 = (2 I - A) C^n = B C^n`
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# The matrix `A` is represented as a `3 x (nx - 1)` matrix as expected by
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# `scipy.linalg.solve_banded` representing the 3 diagonals of the tridiagonal
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# matrix `A`.
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A = np.zeros((3, nx - 1))
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# upper minor diagonal
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A[0, 2:] = -d
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# main diagonal
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A[1, 0] = 1
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A[1, 1:] = 1 + 2 * d
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# and lower minor diagonal
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A[2, :-2] = -d
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A[2, -2] = -2 * d
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# The right hand side matrix `B = 2 I - A` which is represented as a sparse
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# matrix in diag. layout which allows for fast matrix vector multiplication
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B = -A
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B[1, ] += 2
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B = scipy.sparse.spdiags(B, (1, 0, -1), B.shape[1], B.shape[1])
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# set descritized space evaluation points `x` of `C`
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x = np.linspace(0, h, nx)
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# Set initial solution
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C = np.zeros(nx)
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C[0] = 1
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C[-1] = C[-3] # (0 = 0)
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for n in range(nt):
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# generate roughly 400 plots
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if n % (nt // 400) == 0:
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plot(C, x, n * dt, "task01_4")
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# update solution using the implicit schema
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C[:-1] = scipy.linalg.solve_banded((1, 1), A, B @ C[:-1])
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# set/enforce boundary conditions
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C[0] = 1 # left Dirichlet (theoretically not needed)
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C[-1] = C[-3] # right Neumann condition
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# plot final state
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plot(C, x, T, "task01_4")
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# to convert generated image sequence to video use:
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# $> ffmpeg -r 60 -i plots/task01_4_%05d.png -pix_fmt yuv420p video_1_4.mp4
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