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
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
Given an initial condition (u(t=0) = u0) one can follow the time dependence of the temperature field with explicit time evolution method:
Note: Algorithm is stable only when
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 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