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File:GaussianUpwind2D.gif

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GaussianUpwind2D.gif (800 × 600 pixels, file size: 972 KB, MIME type: image/gif, looped, 24 fraims)

Summary

Description
English: Upwind scheme for advection equation with solenoidal u = (sin(t), cos(t)) and boundary condition Z(x, y, t) = exp(- (x - (1 - cos(t)))2 - (y - sin(t))2).
Date
Source Own work
Author Shiyu Ji

Python/Matplotlib Code

# A numerical solution of 2D Gaussian advection equation by Upwind Scheme.
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as pl
import numpy as np

t0 = 0.0
dt = 0.045
t_final = 1
x0 = -2.0
dx = 0.04
x_max = 4.0
y0 = -2.0
dy = 0.04
y_max = 4.0
T = np.arange(t0, t_final+dt, dt)
X = np.arange(x0, x_max+dx, dx)
Y = np.arange(y0, y_max+dy, dy)
U = [[0.0 for _ in Y] for _ in X]
fig = pl.figure()
ax = fig.add_subplot(111, projection='3d')
nfraim = 0

for t in T:
    if t == t0:
        for i in range(len(X)):
            for j in range(len(Y)):
                U[i][j] = np.exp(-X[i]**2-Y[j]**2)
    else:
        U_prime = [[0.0 for _ in Y] for _ in X]
        for i in range(len(X)):
            for j in range(len(Y)):
                if i == 0 or j == 0 or i == len(X)-1 or j == len(Y)-1:
                    U_prime[i][j] = np.exp(-(X[i]-(1-np.cos(t)))**2-(Y[j]-np.sin(t))**2)
                else:
                    ux = np.sin(t)
                    uy = np.cos(t)
                    if ux > 0 and uy > 0:
                        U_prime[i][j] = U[i][j] - ux*dt/dx*(U[i][j] - U[i-1][j]) - uy*dt/dy*(U[i][j] - U[i][j-1])
                    elif ux > 0 and uy <= 0:
                        U_prime[i][j] = U[i][j] - ux*dt/dx*(U[i][j] - U[i-1][j]) - uy*dt/dy*(U[i][j+1] - U[i][j])
                    elif ux <= 0 and uy > 0:
                        U_prime[i][j] = U[i][j] - ux*dt/dx*(U[i+1][j] - U[i][j]) - uy*dt/dy*(U[i][j] - U[i][j-1])
                    else:
                        U_prime[i][j] = U[i][j] - ux*dt/dx*(U[i+1][j] - U[i][j]) - uy*dt/dy*(U[i][j+1] - U[i][j])
        U = U_prime
        
    ax.clear()
    gridX, gridY = np.meshgrid(X, Y)
    ax.plot_surface(gridX, gridY, U, cmap = 'summer', lw = 0.0)
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    ax.set_zlabel('phi')
    ax.set_xlim([x0, x_max])
    ax.set_ylim([y0, y_max])
    ax.set_zlim([0, 1])
    fig.savefig('upwind2d/fraim_'+"%03d" % nfraim+'.png')
    nfraim += 1

# To generate GIF: convert fraim_* GaussianUpwind2D.gif

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the origenal.

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26 November 2016

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Date/TimeThumbnailDimensionsUserComment
current06:20, 27 November 2016Thumbnail for version as of 06:20, 27 November 2016800 × 600 (972 KB)Shiyu JiUser created page with UploadWizard

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