Alan Dodgson

 Alan Dodgson

Mathematics, Heriot-Watt University

Tenure since 2017

Numerical approximation of stochastic Navier-Stokes equationsAlan Dodgson

In applied mathematics, the Navier-Stokes equations are used to simulate the movement of a wide range of fluids. They can simulate scenarios ranging from liquid moving through a container to the air flow around an aircraft’s wing in flight. Usually such equations are impossible to solve by hand, so we rely on the development of numerical methods to accurately find solutions.

There is always an element of uncertainty when simulating real-world systems, and deterministic models can rarely reproduce the exact behaviour observed in real life. As a result, the field of stochastic processes has developed into a prominent area of mathematical research: we can include stochastic terms in models, which evolve randomly over time; the resulting equations can give more accurate representations of the ‘true’ behaviour observed in the real world.

I will investigate the effect of introducing such a stochastic term into a mathematical model, focusing particularly on the Navier-Stokes equations due to their widespread use in applied mathematics. I will also focus on programming numerical ‘finite element’ algorithms to simulate realistic behaviour. In particular, my work will involve calculating the error that will theoretically arise when using such a method, before backing up these claims by running numerical simulations and discussing the results.

Biography

It was during high school that I realised how much I enjoyed studying maths. After winning several awards, including top performance in Advanced Higher Maths and Physics, I felt especially motivated to continue my studies at university. I entered directly into second year at Heriot-Watt University and graduated in 2017 with a first-class MMath degree. In addition, in my final year I was given the IMA prize for excellence in a Mathematics degree course.

In fourth year of university I wrote my dissertation which discussed ‘Discontinuous Galerkin’ methods, a category of numerical methods used to approximate solutions to differential equations. This work was continued in fifth year - I studied the numerical error arising from the use of finite element methods to solve systems involving stochastic terms, in particular the stochastic heat equation.

In the summer between these two dissertations I worked on a six-week summer project, also funded by the Carnegie Trust, which involved using several numerical methods to simulate the movement of vesicles throughout an organism. The results were tested via comparison to real-world data, obtained from previous experiments by the biology department.