Highlights of Carnegie Vacation Scholarships Event 2019
Project Title: Numerical approximation of stochastic Navier-Stokes equations
In applied mathematics, the Navier-Stokes equations are widely considered to be one of the most important mathematical models, used to represent and simulate the movement of a wide range of fluids. In modern mathematics, they are applied to real-world scenarios ranging from measuring liquid moving through a container to calculating the air flow around an aircraft’s wing in flight. Usually these equations are impossible to solve by hand, so we rely on the development of numerical methods to accurately approximate solutions to these problems.
There is always an element of uncertainty when simulating real-world systems, and mathematical models can rarely reproduce the exact behaviour observed in real life. As a result, the field of stochastic processes has developed into one of the most prominent areas of modern mathematical research: we can include stochastic terms in models, which evolve randomly over time; the resulting equations can give far more accurate representations of the ‘true’ behaviour observed in the real world.
The project investigates 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, and also focuses on programming numerical algorithms in order to simulate realistic behaviour. In particular, it involves 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. In doing so, it can be determined how appropriate finite element methods are when tackling stochastic problems.
Awarded: Carnegie PhD Scholarship
University: Heriot-Watt University