add: task 1, subtask 1-3 python reimplementation
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%% eulero forward DIrichlet
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%% Euler forward Dirichlet
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L=pi;
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L = 1; % domain size (in the lecture notes this is denote h)
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T=2;
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T = 2; % time limit (max time)
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f=@(x,t) 0*x.*t;
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f = @(x,t) 0*x.*t; % rhs of the more general equation `u_t - d u_xx = f`
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c1=@(t) 1+0*t;
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c1 = @(t) 1+0*t; % _right_ boundary condition
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c2=@(t) 0*t;
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c2 = @(t) 0*t; % _left_ boundary condition
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u0=@(x) 0*x;
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u0 = @(x) 0*x; % initial values
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D=0.5;
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D = 0.5; % diffusion parameter `d` in `u_t - d u_xx = f`
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%uex = @(x,t) cos(x).*exp(t);
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%uex = @(x,t) cos(x).*exp(t);
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N=10;
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N = 10; % nr. of _space_ discretization points
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K=200;
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K = 200; % nr. of _time_ discretization points
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[x, t, u] = Dirichlet_EA(L, N, T, K, c1, c2, f, u0, D);
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[x, t, u] = Dirichlet_EA(L, N, T, K, c1, c2, f, u0, D);
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% Report stability condition `D Delta T / (Delta x)^2 > 0.5`
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Delta_T = T / K;
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Delta_x = L / N;
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d = D * Delta_T / Delta_x^2;
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fprintf("Stability Condition: 0.5 >= D * Delta_T / Delta_x^2 = %f\n", d)
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if d > 0.5
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fprintf("-> NOT Stable\n")
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else
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fprintf("-> Stable\n")
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end
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figure(1)
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figure(1)
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for ii=1:K+1
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for ii = 1:K+1 % iterates time
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hold on
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plot(x, u(:, ii)');
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plot(x, u(:, ii)');
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xlim([0 L])
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xlim([0 L])
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pause(0.05);
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pause(0.05);
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hold off
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end
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end
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% 3D plot of space solution over time
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space = linspace(0,L,101);
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space = linspace(0,L,101);
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time = linspace(0,T,201);
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time = linspace(0,T,201);
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[xx,yy] = meshgrid(time,space);
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[xx,yy] = meshgrid(time,space);
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@ -0,0 +1,63 @@
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# Task 1.1, 1.2
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import numpy as np
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from typing import Callable
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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 = 2e6 # solution end time
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nx = 50 # nr of space discretization points
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nt = 20000 # 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 - 1) # time step size
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d = dt * D / dx**2 # stability/stepsize coefficient
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# report stability
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if d > 0.5:
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print("NOT Stable")
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else:
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print("Stable")
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# explicit scheme integration for `u_t - D u_xx = 0` with boundary conditions
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# enforced by `set_bounds` and initial conditions `initial`.
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def integrate(*, name: str, initial: np.array, set_bounds: Callable[[np.array], None]) -> None:
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C = initial
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# Setup boundary conditions
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set_bounds(C)
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i = 0 # index for plot generation
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plt.figure(figsize = (8, 6), dpi = 100)
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for t in range(nt):
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# every 400'th time step save a plot
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if t % (nt // 400) == 0:
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plt.clf()
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plt.plot(np.linspace(0, h, nx), C)
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plt.xlim([0, h])
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plt.ylim([0, 1.2])
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plt.savefig(f"plots/{name}_{i:0>5}.png")
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i += 1
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# update solution using the explicit schema
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C[1:-1] += d * (C[2:] - 2 * C[1:-1] + C[:-2])
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# update right Neumann BC
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set_bounds(C)
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# Subtask 1 boundary conditions (Dirichlet and Neumann)
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def bounds_1(C):
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C[0] = 1
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C[-1] = C[-2]
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# Subtask 2 boundary conditions (two Dirichlet)
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def bounds_2(C):
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C[0] = 1
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C[-1] = 0
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# run simulations
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integrate(name = 'task01_1', initial = np.zeros(nx), set_bounds = bounds_1)
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integrate(name = 'task01_2', initial = np.zeros(nx), set_bounds = bounds_2)
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# to convert generated image sequence to video use:
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# $> ffmpeg -r 60 -i plots/task01_1_%05d.png -pix_fmt yuv420p video_1_1.mp4
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# $> ffmpeg -r 60 -i plots/task01_2_%05d.png -pix_fmt yuv420p video_1_2.mp4
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@ -0,0 +1,50 @@
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# Task 1.3
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import numpy as np
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from typing import Callable
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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 = 2e6 # solution end time
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nx = 50 # nr of space discretization points
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nt = 20000 # 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 - 1) # time step size
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d = dt * D / dx**2 # stability/stepsize coefficient
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# Setup implicit scheme equation matrix
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T = (1 + 2 * d) * np.eye(nx) - d * np.eye(nx, k = 1) - d * np.eye(nx, k = -1)
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# fix boundary condition equations
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T[0, 0] = 1 # Left Dirichlet BC
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T[0, 1] = 0
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T[-1, -2] = 1 # Right Neumann BC
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T[-1, -1] = 0
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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[-2] # (0 = 0)
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i = 0 # index for plot generation
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plt.figure(figsize = (8, 6), dpi = 100)
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for t in range(nt):
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# every 400'th time step save a plot
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if t % (nt // 400) == 0:
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plt.clf()
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plt.plot(np.linspace(0, h, nx), C)
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plt.xlim([0, h])
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plt.ylim([0, 1.2])
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plt.savefig(f"plots/task01_3_{i:0>5}.png")
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i += 1
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# update solution using the explicit schema
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C = np.linalg.solve(T, C)
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# fix BC conditions (theoretically, they are set by the update but for
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# stability reasons (numerical) we enforce the correct values)
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C[0] = 1
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C[-1] = C[-2]
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# to convert generated image sequence to video use:
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# $> ffmpeg -r 60 -i plots/task01_3_%05d.png -pix_fmt yuv420p video_1_3.mp4
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