Abgabe

Exercise_03/abgabe/task01.py

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

Exercise_03/abgabe/task02.py

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+import numpy as np
+from scipy.sparse import spdiags
+from scipy.linalg import solve_banded
+from matplotlib import pyplot as plt
+
+# Config
+dx = 0.01           # space step size
+h = 1.5             # space domain (max x size)
+
+# plots a solution to file, simply for code size reduction in `solve`
+def plot(x, t, C, C0, analytic, plotname):
+    plt.clf()
+    plt.plot(x, C, "o")
+    plt.xlim([0, h])
+    plt.title(plotname)
+    plt.xlabel("Periodic Space: x")
+    plt.ylabel("Wave Form: C(x, t)")
+    if analytic is not None:
+        plt.plot(x, analytic(x, t, C0))
+    plt.savefig(f"plots/{plotname}_{plot.i:0>5}.png")
+    plot.i += 1
+plot.i = 0
+
+# solver for `C_t + U C_x = 0` given initial state `C`, the current `C0` which
+# incorporates `U` as well as the space and time discretization and the number
+# of time steps `nt`.
+def solve(C, C0, nt, plotname = None, plotskip = 1, analytic = None):
+    nx = C.shape[0]
+    if plotname is not None:
+        plot.i = 0                                      # (re)set plot counter
+        x = np.linspace(0, h, nx)                       # space grid points
+        fig = plt.figure(figsize = (8, 6), dpi = 100)   # setup plot figure size
+        plot(x, 0, C, C0, analytic, plotname)           # plot initial solution
+    # space grid point `Indices Minus 1`
+    im1 = np.array([nx - 1] + list(range(nx - 1)))
+    for t in range(nt):
+        # upwind update scheme
+        C = (1 - C0) * C + C0 * C[im1]
+        if plotname is not None and ((t + 1) % plotskip == 0):
+            plot(x, t + 1, C, C0, analytic, plotname)    # plot current solution
+    return C
+
+# setup initial conditions
+x = np.linspace(0, h, int(h / dx))
+gauss = np.exp(-10 * (4 * x - 1)**2)
+square = np.array((0.1 < x) * (x < 0.3), dtype = "float")
+
+def gauss_analytic(x, t, C0):
+    return np.exp(-10 * (4 * (np.mod(x - t * dx / C0, h + 1e-9)) - 1)**2)
+def square_analytic(x, t, C0):
+    x = x - t * dx / C0
+    x = np.mod(x, h + 1e-9) # hack to avoid shift in the plot
+    return np.array((0.1 < x) * (x < 0.3), dtype = "float")
+
+solve(gauss, 1, 200, "task02_gauss_C0_1", analytic = gauss_analytic)
+solve(square, 1, 200, "task02_square_C0_1", analytic = square_analytic)
+solve(gauss, 0.7, 200, "task02_gauss_C0_0.7", analytic = gauss_analytic)
+solve(square, 0.7, 200, "task02_square_C0_0.7", analytic = square_analytic)
+solve(gauss, 1.1, 200, "task02_gauss_C0_1.1", analytic = gauss_analytic)
+solve(square, 1.1, 200, "task02_square_C0_1.1", analytic = square_analytic)
+
+# To generate videos out of the generated plots
+# $> ffmpeg -y -r 60 -i plots/task02_gauss_C0_1_%05d.png -pix_fmt yuv420p task02_gauss_C0_1.mp4
+# $> ffmpeg -y -r 60 -i plots/task02_square_C0_1_%05d.png -pix_fmt yuv420p task02_square_C0_1.mp4
+# $> ffmpeg -y -r 60 -i plots/task02_gauss_C0_0.7_%05d.png -pix_fmt yuv420p task02_gauss_C0_0.7.mp4
+# $> ffmpeg -y -r 60 -i plots/task02_square_C0_0.7_%05d.png -pix_fmt yuv420p task02_square_C0_0.7.mp4
+# $> ffmpeg -y -r 60 -i plots/task02_gauss_C0_1.1_%05d.png -pix_fmt yuv420p task02_gauss_C0_1.1.mp4
+# $> ffmpeg -y -r 60 -i plots/task02_square_C0_1.1_%05d.png -pix_fmt yuv420p task02_square_C0_1.1.mp4