Assignment

Author: jussienko

numpy/heat-equation/README.md

@@ -0,0 +1,39 @@
+## Two dimension hear equation
+
+Heat (or diffusion) equation is
+
+<!--- Equation
+\frac{\partial u}{\partial t} = \alpha \nabla^2 u
+---> 
+
+![img](http://quicklatex.com/cache3/d2/ql_b3f6b8bdc3a8862c73c5a97862afb9d2_l3.png)
+
+where **u(x, y, t)** is the temperature field that varies in space and time, and α is thermal diffusivity constant. The two dimensional Laplacian can be discretized with finite differences as
+
+<!--- Equation
+\begin{align*}
+\nabla^2 u  &= \frac{u(i-1,j)-2u(i,j)+u(i+1,j)}{(\Delta x)^2} \\
+ &+ \frac{u(i,j-1)-2u(i,j)+u(i,j+1)}{(\Delta y)^2}
+\end{align*}
+---> 
+
+![img](http://quicklatex.com/cache3/2d/ql_59f49ed64dbbe76704e0679b8ad7c22d_l3.png)
+
+Given an initial condition (u(t=0) = u0) one can follow the time dependence of the temperature field with explicit time evolution method:
+
+<!--- Equation
+u^{m+1}(i,j) = u^m(i,j) + \Delta t \alpha \nabla^2 u^m(i,j) 
+---> 
+
+![img](http://quicklatex.com/cache3/9e/ql_9eb7ce5f3d5eccd6cfc1ff5638bf199e_l3.png)
+
+Note: Algorithm is stable only when
+
+<!--- Equation
+\Delta t < \frac{1}{2 \alpha} \frac{(\Delta x \Delta y)^2}{(\Delta x)^2 (\Delta y)^2}
+--->
+
+![img](http://quicklatex.com/cache3/d1/ql_0e7107049c9183d11dbb1e81174280d1_l3.png)
+
+Implement two dimensional heat equation with NumPy using the initial temperature field in the file [bottle.dat](bottle.dat) (the file consists of a header and 200 x 200 data array). As a boundary condition use fixed values as given in the initial field. You can start from the skeleton code in the file [skeleton.py](skeleton.py).
+