Thomas Russell: Programmer & Designer
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Solution to the Poisson Equation in C++ using Successive Over-relaxation

Poisson Equation Plots

Note: The first section of this blog post is mathematical and useful to understand, however, if you are just looking for an implementation you can skip through to the first code block.

Due to the interest in my previous blog post: Solution to the Laplace Equation in C++ using successive over-relaxation, I have decided to write a follow-up post on how to extend the method to allow for solutions to the Poisson Equation.

The Poisson Equation is the general case of the Laplace Equation and takes the following form:

$$\nabla^{2} \phi = f(\vec{r})$$

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Solution to the Laplace Equation in C++ using successive over-relaxation

Note: The first section of this blog post is mathematical and useful to understand, however, if you are just looking for an implementation you can skip through to the first code block.

Partial Differential Equations are notoriously difficult to solve analytically; in all but the simplest cases, there often does not exist a solution in elementary form. This would be acceptable, but for the ubiquity of partial differential equations in physical models. So Mathematicians and Computer Scientists had to develop methods of solving PDEs numerically.

In this blog post, I will show one such method of solving Laplace’s equation (a type of PDE) in $\mathbb{R}^{2}$ using the method of successive over-relaxation, an iterative technique which involves splitting the relevant domain into a grid and sampling each point in a fixed order. For those who don’t know, or have forgotten, Laplace’s equation is written as follows:

$$\nabla^{2} \psi = 0$$

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Musings on the Technological Singularity

Recently there has been increased discussion in the media about the so-called Technological Singularity (for instance this BBC article on Stephen Hawking’s opinion). For those who are unfamiliar, this is a hypothetical event whereby humans create a machine that is capable of producing a machine more capable than itself, which will also be capable of producing a machine more capable than itself, and so on ad infinitum; resulting in the development of a machine that no longer requires the existence of human beings and thus eliminates the entirety of the human race (or alternatively enslaves us in a Matrix-style coup). Read more »

OCR C3 & C4 Revision Guide

I’m currently in the process of applying to internships, and as I was digging around my Google Drive (which is what I’m using to draft my cover letters) I came across a revision guide I made a couple of years ago to help people with OCR A-level maths revision (specifically the C3 and C4 sections).

I’m sure that there are bits missing, but it covers 5 basic topics:

  1. Lines and basic linear algebra
  2. Differentiation and Integration
  3. Numerical Methods: Root finding & Quadrature methods
  4. Algebra and Functions
  5. Trigonometry

I just thought I’d release it here to see if it can possibly help anyone! (N.B: The wide margins are for notes to self whilst reading).

Download Link: C3 & C4 Revision Guide

Added comments!!

I’ve finally got around to adding comments to the blog, so now you can leave your thoughts/suggestions/questions at the bottom of the relevant blog post rather than having to email me manually!!

I’m still not entirely satisfied with the design or functionality, and for some reason one of the scrips is not loading as expected (for those of you familiar with WordPress it’s comment-reply.js), but I’m hoping to fix that in the next couple of days.

Hopefully I’ll hear some feedback from people in the comments in the near future (maybe when I get some more traffic haha!)

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