From cc3fa78db6ef69fc11d0b348752a992d892760cf Mon Sep 17 00:00:00 2001 From: daniel Date: Fri, 13 May 2022 18:52:15 +0200 Subject: [PATCH] add: task 1, subtask 1-3 python reimplementation --- Exercise_03/NSSC_1.m | 56 ++++++++++++++++++++------------- Exercise_03/task01_1-2.py | 63 ++++++++++++++++++++++++++++++++++++++ Exercise_03/task01_3.py | 50 ++++++++++++++++++++++++++++++ Exercise_03/video_1_1.mp4 | Bin 0 -> 74867 bytes Exercise_03/video_1_2.mp4 | Bin 0 -> 55043 bytes Exercise_03/video_1_3.mp4 | Bin 0 -> 58344 bytes 6 files changed, 148 insertions(+), 21 deletions(-) create mode 100644 Exercise_03/task01_1-2.py create mode 100644 Exercise_03/task01_3.py create mode 100644 Exercise_03/video_1_1.mp4 create mode 100644 Exercise_03/video_1_2.mp4 create mode 100644 Exercise_03/video_1_3.mp4 diff --git a/Exercise_03/NSSC_1.m b/Exercise_03/NSSC_1.m index f548749..e694ce7 100755 --- a/Exercise_03/NSSC_1.m +++ b/Exercise_03/NSSC_1.m @@ -1,28 +1,42 @@ -%% eulero forward DIrichlet -L=pi; -T=2; -f=@(x,t) 0*x.*t; -c1=@(t) 1+0*t; -c2=@(t) 0*t; -u0=@(x) 0*x; -D=0.5; -%uex=@(x,t) cos(x).*exp(t); +%% Euler forward Dirichlet +L = 1; % domain size (in the lecture notes this is denote h) +T = 2; % time limit (max time) +f = @(x,t) 0*x.*t; % rhs of the more general equation `u_t - d u_xx = f` +c1 = @(t) 1+0*t; % _right_ boundary condition +c2 = @(t) 0*t; % _left_ boundary condition +u0 = @(x) 0*x; % initial values +D = 0.5; % diffusion parameter `d` in `u_t - d u_xx = f` +%uex = @(x,t) cos(x).*exp(t); -N=10; -K=200; -[x,t,u]=Dirichlet_EA(L,N,T,K,c1,c2,f,u0,D); +N = 10; % nr. of _space_ discretization points +K = 200; % nr. of _time_ discretization points +[x, t, u] = Dirichlet_EA(L, N, T, K, c1, c2, f, u0, D); -figure(1) -for ii=1:K+1 - plot(x,u(:,ii)'); - xlim([0 L]) - pause(0.05); +% Report stability condition `D Delta T / (Delta x)^2 > 0.5` +Delta_T = T / K; +Delta_x = L / N; +d = D * Delta_T / Delta_x^2; +fprintf("Stability Condition: 0.5 >= D * Delta_T / Delta_x^2 = %f\n", d) +if d > 0.5 + fprintf("-> NOT Stable\n") +else + fprintf("-> Stable\n") end -space=linspace(0,L,101); -time=linspace(0,T,201); -[xx,yy]=meshgrid(time,space); -%exsol=uex(yy,xx); +figure(1) +for ii = 1:K+1 % iterates time + hold on + plot(x, u(:, ii)'); + xlim([0 L]) + pause(0.05); + hold off +end + +% 3D plot of space solution over time +space = linspace(0,L,101); +time = linspace(0,T,201); +[xx,yy] = meshgrid(time,space); +%exsol = uex(yy,xx); figure(2) mesh(t,x,u) %figure(2) diff --git a/Exercise_03/task01_1-2.py b/Exercise_03/task01_1-2.py new file mode 100644 index 0000000..c6ad816 --- /dev/null +++ b/Exercise_03/task01_1-2.py @@ -0,0 +1,63 @@ +# Task 1.1, 1.2 +import numpy as np +from typing import Callable +from matplotlib import pyplot as plt + +# Config +D = 1e-6 # diffusion coefficient +h = 1 # space domain (max x size) +T = 2e6 # solution end time +nx = 50 # nr of space discretization points +nt = 20000 # nr of time discretization points + +# derived constants +dx = h / (nx - 1) # space step size +dt = T / (nt - 1) # time step size +d = dt * D / dx**2 # stability/stepsize coefficient + +# report stability +if d > 0.5: + print("NOT Stable") +else: + print("Stable") + +# explicit scheme integration for `u_t - D u_xx = 0` with boundary conditions +# enforced by `set_bounds` and initial conditions `initial`. +def integrate(*, name: str, initial: np.array, set_bounds: Callable[[np.array], None]) -> None: + C = initial + # Setup boundary conditions + set_bounds(C) + + i = 0 # index for plot generation + plt.figure(figsize = (8, 6), dpi = 100) + for t in range(nt): + # every 400'th time step save a plot + if t % (nt // 400) == 0: + plt.clf() + plt.plot(np.linspace(0, h, nx), C) + plt.xlim([0, h]) + plt.ylim([0, 1.2]) + plt.savefig(f"plots/{name}_{i:0>5}.png") + i += 1 + # update solution using the explicit schema + C[1:-1] += d * (C[2:] - 2 * C[1:-1] + C[:-2]) + # update right Neumann BC + set_bounds(C) + +# Subtask 1 boundary conditions (Dirichlet and Neumann) +def bounds_1(C): + C[0] = 1 + C[-1] = C[-2] + +# Subtask 2 boundary conditions (two Dirichlet) +def bounds_2(C): + C[0] = 1 + C[-1] = 0 + +# run simulations +integrate(name = 'task01_1', initial = np.zeros(nx), set_bounds = bounds_1) +integrate(name = 'task01_2', initial = np.zeros(nx), set_bounds = bounds_2) + +# to convert generated image sequence to video use: +# $> ffmpeg -r 60 -i plots/task01_1_%05d.png -pix_fmt yuv420p video_1_1.mp4 +# $> ffmpeg -r 60 -i plots/task01_2_%05d.png -pix_fmt yuv420p video_1_2.mp4 diff --git a/Exercise_03/task01_3.py b/Exercise_03/task01_3.py new file mode 100644 index 0000000..b824a47 --- /dev/null +++ b/Exercise_03/task01_3.py @@ -0,0 +1,50 @@ +# Task 1.3 +import numpy as np +from typing import Callable +from matplotlib import pyplot as plt + +# Config +D = 1e-6 # diffusion coefficient +h = 1 # space domain (max x size) +T = 2e6 # solution end time +nx = 50 # nr of space discretization points +nt = 20000 # nr of time discretization points + +# derived constants +dx = h / (nx - 1) # space step size +dt = T / (nt - 1) # time step size +d = dt * D / dx**2 # stability/stepsize coefficient + +# Setup implicit scheme equation matrix +T = (1 + 2 * d) * np.eye(nx) - d * np.eye(nx, k = 1) - d * np.eye(nx, k = -1) +# fix boundary condition equations +T[0, 0] = 1 # Left Dirichlet BC +T[0, 1] = 0 +T[-1, -2] = 1 # Right Neumann BC +T[-1, -1] = 0 + +# Set initial solution +C = np.zeros(nx) +C[0] = 1 +C[-1] = C[-2] # (0 = 0) + +i = 0 # index for plot generation +plt.figure(figsize = (8, 6), dpi = 100) +for t in range(nt): + # every 400'th time step save a plot + if t % (nt // 400) == 0: + plt.clf() + plt.plot(np.linspace(0, h, nx), C) + plt.xlim([0, h]) + plt.ylim([0, 1.2]) + plt.savefig(f"plots/task01_3_{i:0>5}.png") + i += 1 + # update solution using the explicit schema + C = np.linalg.solve(T, C) + # fix BC conditions (theoretically, they are set by the update but for + # stability reasons (numerical) we enforce the correct values) + 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