The two-dimensional diffusion problem in parallel

This Fortran program solves the unsteady, two-dimensional temperature diffusion equation.

$$ \frac{\partial T}{\partial t} = D \nabla^2T = D \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) $$

Here T is the temperature to be solved and D the diffusion constant.

Problem definition

The chosen domain is the unit square, such that $(L_x, L_y) = (1, 1)$. The initial condition $T(x, y, t=0) = 0$ is used.

Dirichlet boundary conditions are used to model an unsteady diffusion of heat from the bounds of the unit square towards the origin:

$$ T(x, 0, t) = T(x, L_y, t) = T(0, y, t) = T(L_x, y, t) = 1. $$

Discretisation

A uniform Cartesian grid is chosen for simplicity. The grid sizes are expressed below.

$$ \Delta x = \frac{L_x}{N_x - 1} \qquad \Delta y = \frac{L_y}{N_y - 1} $$

An explicit Eulerian time integration produces the centered algorithm integrated in this Fortran code.

$$ \frac{T_{i,j}^{n+1} - T_{i,j}^n}{\Delta t} = D \left( \frac{T_{i+1, j}^n - 2 T^n_{i,j} + T^n_{i-1,j}}{\Delta x^2} + \frac{T_{i, j+1}^n - 2 T^n_{i,j} + T^n_{i,j-1}}{\Delta y^2} \right) $$

This program uses $D = 1$ as the default.

Features

Input data (domain information, diffusion constant and similar) can be read in from a file input_data.dat or manually defined in the module inputs.f90.

The CPU and wall lock time can be measured using the timings routines included.

This program will use MPI for parallelisation. Exchange of domain boundaries between communicators is handled by module_bounds.f90.

Included is a basic python visualisation script for providing insights from timing data.

Execution

To run the code:

make new
mpirun -n <number_of_processes> ./diffusion

---

Credits

Developed by Elloïse Fangel-Lloyd, Dana Lüdemann and Clemens Zengler.