The two-dimensional diffusion problem

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.

$$ \Delta x = \frac{L_x}{N_x - 1} \hspace{6cm} \Delta y = \frac{L_y}{N_y - 1} $$

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.

Execution

To run the code:

make new
./diffusion 

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Credits

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