add: task 1, subtask 1-3 python reimplementation

This commit is contained in:
Daniel Kapla 2022-05-13 18:52:15 +02:00
parent dad2f0151f
commit cc3fa78db6
6 changed files with 148 additions and 21 deletions

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%% eulero forward DIrichlet %% Euler forward Dirichlet
L=pi; L = 1; % domain size (in the lecture notes this is denote h)
T=2; T = 2; % time limit (max time)
f=@(x,t) 0*x.*t; f = @(x,t) 0*x.*t; % rhs of the more general equation `u_t - d u_xx = f`
c1=@(t) 1+0*t; c1 = @(t) 1+0*t; % _right_ boundary condition
c2=@(t) 0*t; c2 = @(t) 0*t; % _left_ boundary condition
u0=@(x) 0*x; u0 = @(x) 0*x; % initial values
D=0.5; D = 0.5; % diffusion parameter `d` in `u_t - d u_xx = f`
%uex=@(x,t) cos(x).*exp(t); %uex = @(x,t) cos(x).*exp(t);
N=10; N = 10; % nr. of _space_ discretization points
K=200; K = 200; % nr. of _time_ discretization points
[x,t,u]=Dirichlet_EA(L,N,T,K,c1,c2,f,u0,D); [x, t, u] = Dirichlet_EA(L, N, T, K, c1, c2, f, u0, D);
figure(1) % Report stability condition `D Delta T / (Delta x)^2 > 0.5`
for ii=1:K+1 Delta_T = T / K;
plot(x,u(:,ii)'); Delta_x = L / N;
xlim([0 L]) d = D * Delta_T / Delta_x^2;
pause(0.05); 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 end
space=linspace(0,L,101); figure(1)
time=linspace(0,T,201); for ii = 1:K+1 % iterates time
[xx,yy]=meshgrid(time,space); hold on
%exsol=uex(yy,xx); 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) figure(2)
mesh(t,x,u) mesh(t,x,u)
%figure(2) %figure(2)

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Exercise_03/task01_1-2.py Normal file
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# 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

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Exercise_03/task01_3.py Normal file
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# 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)
C[0] = 1
C[-1] = C[-2]
# to convert generated image sequence to video use:
# $> ffmpeg -r 60 -i plots/task01_3_%05d.png -pix_fmt yuv420p video_1_3.mp4

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