155 lines
5.0 KiB
Python
155 lines
5.0 KiB
Python
import numpy as np
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from scipy.sparse import spdiags
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from scipy.linalg import solve_banded
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from matplotlib import pyplot as plt
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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 = 2e5 # solution end time
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nx = 50 # nr of space discretization points
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nt = 1000 # nr of time discretization 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 # time step size
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d = dt * D / dx**2 # stability/stepsize coefficient
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# Task 1.1 equation system matrix `C^n+1 = A1 C^n`
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def A1(d):
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a1 = np.zeros([3, nx])
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# superdiagonal (above the main diagonal)
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a1[0, 1] = 0
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a1[0, 2:] = d
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# main diagonal
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a1[1, 0] = 1
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a1[1, 1:] = 1 - 2 * d
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# subdiagonal (below the main diagonal)
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a1[2, :-2] = d
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a1[2, -2] = 2 * d
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# convert to sparse tridiagonal matrix
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return spdiags(a1, [1, 0, -1], nx, nx)
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# Task 1.2 equation system matrix `C^n+1 = A2 C^n`
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def A2(d):
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a2 = A1(d).data
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# addapt boundary condition coefficients, the rest is identical
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a2[0, -1] = a2[1, -1] = a2[2, -2] = 0
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# convert to sparse tridiag
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return spdiags(a2, [1, 0, -1], nx, nx)
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# Task 1.3 equation system matrix for update rule `A3 C^n+1 = C^n`
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def A3(d):
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a3 = np.zeros([3, nx])
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# superdiagonal
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a3[0, 1] = 0
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a3[0, 2:] = -d
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# main diagonal
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a3[1, 0] = 1
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a3[1, 1:] = 1 + 2 * d
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# supdiagonal
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a3[2, :-2] = -d
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a3[2, -2] = -2 * d
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# as tridiag
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return spdiags(a3, [1, 0, -1], nx, nx)
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# Task 1.4 equation system matrix `A4`` in `(I + A4) C^n+1 = (I - A4) C^n`
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def A4(d):
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a4 = np.zeros([3, nx])
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# superdiagonal
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a4[0, 2:] = -d / 2
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# main diagonal
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a4[1, 1:] = d
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# supdiagonal
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a4[2, :-2] = -d / 2
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a4[2, -2] = -d
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return spdiags(a4, [1, 0, -1], nx, nx)
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# setup `I + A4` and `I - A4`
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def IpA4(d):
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ipA4 = A4(d).data
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ipA4[1, ] += 1
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return spdiags(ipA4, [1, 0, -1], nx, nx)
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def ImA4(d):
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imA4 = -A4(d).data
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imA4[1, ] += 1
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return spdiags(imA4, [1, 0, -1], nx, nx)
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# plots a solution to file, simply for code size reduction in `solve`
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def plot(x, C, name):
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plt.clf()
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plt.plot(x, C)
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plt.xlim([0, h])
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plt.ylim([0, 1.2])
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plt.title(name)
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plt.xlabel("Distance from Source: x")
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plt.ylabel("Concentration: C(x, t)")
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plt.savefig(f"plots/{name}_{plot.i:0>5}.png")
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plot.i += 1
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plot.i = 0
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# solver for `C_t - D C_xx = 0` given discretized equation system matrices for
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# a time update rule `L C^n+1 = R C^n` starting from an initial solution `C`.
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# The update is iterated for `nt` time steps corresponding to a solution at
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# time `T = dt nt`
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def solve(C, L, R, nt, plotname = None, plotskip = 5):
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if plotname is not None:
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plot.i = 0 # (re)set plot counter
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x = np.linspace(0, h, nx) # space grid points
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fig = plt.figure(figsize = (8, 6), dpi = 100) # setup plot figure size
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plot(x, C, plotname) # plot initial solution
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for t in range(nt):
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rhs = C if R is None else R @ C
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C = rhs if L is None else solve_banded((1, 1), L.data, rhs)
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if plotname is not None and ((t + 1) % plotskip == 0):
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plot(x, C, plotname) # plot current solution
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return C
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# setup initial solution
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C0 = np.zeros(nx)
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C0[0] = 1
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# run simulation and generate plots for each subtask
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C1 = solve(C0, None, A1(d), nt, "task01_1")
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Cx = solve(C0, None, A1(0.55), 200, "task01_x", 1) # run with unstable `d > 0.5`
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C2 = solve(C0, None, A2(d), nt, "task01_2")
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C3 = solve(C0, A3(d), None, nt, "task01_3")
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C4 = solve(C0, IpA4(d), ImA4(d), nt, "task01_4")
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# to convert generated image sequence to video use:
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# $> ffmpeg -y -r 60 -i plots/task01_1_%05d.png -pix_fmt yuv420p video_1_1.mp4
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# $> ffmpeg -y -r 60 -i plots/task01_x_%05d.png -pix_fmt yuv420p video_1_x.mp4
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# $> ffmpeg -y -r 60 -i plots/task01_2_%05d.png -pix_fmt yuv420p video_1_2.mp4
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# $> ffmpeg -y -r 60 -i plots/task01_3_%05d.png -pix_fmt yuv420p video_1_3.mp4
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# $> ffmpeg -y -r 60 -i plots/task01_4_%05d.png -pix_fmt yuv420p video_1_4.mp4
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# compute till no time change, e.g. `t -> inf`
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C1inf = solve(C0, None, A1(d), 20000)
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C2inf = solve(C0, None, A2(d), 20000)
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x = np.linspace(0, h, nx) # space grid points
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fig = plt.figure(figsize = (8, 6), dpi = 100) # setup plot figure size
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plot.i = 0 # reset plot counter
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plot(x, C1inf, "Cinf")
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plot(x, C2inf, "Cinf")
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# # All plots in one
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C1 = solve(C0, None, A1(d), nt)
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C2 = solve(C0, None, A2(d), nt)
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C3 = solve(C0, A3(d), None, nt)
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C4 = solve(C0, IpA4(d), ImA4(d), nt)
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x = np.linspace(0, h, nx) # space grid points
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plt.figure(figsize = (8, 6), dpi = 100) # setup plot figure size
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plt.plot(x, C1, label = "1.1")
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plt.plot(x, C2, label = "1.2")
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plt.plot(x, C3, label = "1.3")
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plt.plot(x, C4, label = "1.4")
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plt.xlim([0, h])
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plt.ylim([0, 1.2])
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plt.title("All")
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plt.legend()
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plt.xlabel("Distance from Source: x")
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plt.ylabel("Concentration: C(x, t)")
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plt.savefig(f"plots/task01_all.png")
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