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Two dimensional heat equation
=============================
This folder contains a code which solves two dimensional heat equation
with OpenMP parallelization. A coarse grained parallelization is used, where
the threads are launched in the start of the program and kept alive throughout
the whole execution.
Heat (or diffusion) equation is
<!-- Equation
\frac{\partial u}{\partial t} = \alpha \nabla^2 u
-->
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*}
-->
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)
-->
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}
-->
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.
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 code can be run with arbitrary number of OpenMP threads. The default
initial temperature field is a disk in a 2000 x 2000 grid. 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: OMP_NUM_THREADS=4 ./heat_omp
* Initial field from a file: OMP_NUM_THREADS=4 ./heat_omp bottle.dat
* Initial field from a file, given number of time steps:
OMP_NUM_THREADS=4./heat_omp bottle.dat 1000
* Default pattern with given dimensions and time steps:
OMP_NUM_THREADS=4 ./heat_omp 800 800 1000
The program produces a series of heat_XXXX.png files which show the
time development of the temperature field