NumericalDiffusionScheme

Overview

This repository contains some Python (2.7) code which may be used to solve the one-dimensional diffusion equation with variable diffusivity k(x) and non-zero source term S(x, t), subject to Neumann boundary conditions:

with

where 0 < x < L.

The Documentation sub-directory contains information on the numerical scheme and how it is derived and set up. The code has been written to match the symbols in the document "Implementation_Notes.pdf" as closely as possible.

Usage

In the code sub-directory are three python files:

• diffusion_scheme.py
• plotting.py
• test_me.py

The actual numerical scheme is contained within two functions in diffusion_scheme.py. test_me.py can be run directly from the command line set in the top (repository) directory using

python -i Code\test_me.py

which will generate a figure (using the sub-routines in plotting.py) showing the results of the numerical integration on a specific test case (constant k, zero S - the reason for this is so that an analytic solution can be used for comparison - see the documentation for more details). It can be run again with different parameters by then using the command main(L=4.0, nt=15), for example. Type help(main) to see which other parameters may be changed.

Also shown is a plot which demonstrates that the numerical scheme is conserving q - since Neumann boundary conditions are used and the scheme is a finite volume method, the total integral quantity of q(x) over the domain should remain constant as it is integrated forwards in time. The scheme has been verified to conserve q for (i) constant diffusivity k and (ii) position-dependent diffusivity k(x) (18/04/2018). Unfortunately this numerical scheme does not appear to conserve q when the source S is time dependent (although the solutions retain stability (and probably accuracy depending on the nature of the time dependency)); it is possible this may be due to the back-end matrix inversion methods (i.e. potentially solvable) but this is not being investigated further at this time.

Dependencies

• Python 2.7.14
• MatPlotLib 2.2.2
• NumPy 1.14.3