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Solving the advection-diffusion equation on an non-uniform mesh with the finite-volume method

This repo is basically my notes on learning the finite-volume method when applied to the advection-diffusion equation. The methods are based on the the book by W. Hundsdorfer and J. G. Verwer, Numerical solutions of time-dependent advection-diffusion reaction equations.

For my lab-book notes on solving the advection-diffusion equation with the finite-volume method see the accompanying gh-pages branch, http://danieljfarrell.github.io/FVM/index.html.

The code is written in python using numpy as scipy libraries.

Ideas I wanted to explore:

• How to implement a cell centred mesh?
• How to implement a non-uniform mesh?
• How to include adaptive upwinding?

• This means that the discretisation automatically adjusts to an appropriate scheme depending on the local value of the Peclet number.
• I use exponentially fitting which is a nice method because it enables to equations to be solved for any Peclet number.
• A central difference scheme is limited to Peclet number < 2, however a upwind scheme the Peclet number is unbounded.
• Please not that the CFL condition imposes a limited on the time step when using upwind method.

• How to include Robin boundary conditions which do not allow any out flow.
• How to include Dirichlet boundary conditions.
• How to export a movie of time-dependent simulations.