Describe the heat equation in a bit more detail

heat-equation/README.md

+Two dimensional heat equation
+=============================
+This folder contains a code which solves two dimensional heat equation
+with MPI parallelization. The code features non-blocking point-to-point 
+communication, user defined datatypes, collective communication, 
+and parallel I/O with MPI I/O.
+
+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)
+
+The two dimensional grid is decomposed along both dimensions, and the
+communication of boundary data is overlapped with computation. Restart files
+are written and read with MPI I/O.
+
+Compilation instructions
+------------------------
+For building and running the example one needs to have the
+[libpng](http://www.libpng.org/pub/png/libpng.html) library installed. In
+addition, working MPI environment is required. For Python version mpi4py and
+matplotlib are needed.
+
+Move to proper subfolder (C or Fortran) and modify the top of the **Makefile**
+according to your environment (proper compiler commands and compiler flags).
+Code can be build simple with **make**
+
+How to run
+----------
+The number of MPI ranks has to be a factor of the grid dimension (default 
+dimension is 200). The default initial temperature field is a disk. Initial
+temperature field can be read also from a file, the provided **bottle.dat** 
+illustrates what happens to a cold soda bottle in sauna.
+
+
+ * Running with defaults: mpirun -np 4 ./heat_mpi
+ * Initial field from a file: mpirun -np 4 ./heat_mpi bottle.dat
+ * Initial field from a file, given number of time steps:
+   mpirun -np 4 ./heat_mpi bottle.dat 1000
+ * Defauls pattern with given dimensions and time steps:
+   mpirun -np 4 ./heat_mpi 800 800 1000
+
+  The program produces a series of heat_XXXX.png files which show the
+  time development of the temperature field
+