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 -->  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. 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