Research
My research interests lie primarily in the application of mathematics to scientific problems in physics and engineering. I have developed expertise in numerical partial differential equations (PDEs), stability analysis, domain decomposition methods, compact algorithm methods, multiscale methods, operator splitting methods, and mathematical modeling. I am particularly interested in studying PDEs that model highly oscillatory problems, such as Helmholtz equations.
Below are some of the topics explored in my research.
If you'd like to know more, please see my List of Publications and Research Statement.
Stability Analysis
Stability is a crucial property required for the study of numerical PDEs. I explore two main types of stability. In conventional stability analysis we focus on the step size of the algorithm. In cases where the algorithm is unstable in a conventional sense, it can be interesting to consider non-conventional stability analysis which emphasises the influence of certain physical parameters.
Highly Oscillatory PDE Simulations
Optical beams are often modeled by radially symmetric initial-boundary value problems, such as a Helmholtz equation with a Gaussian beam initial function. These problems are well-suited for asymptotic stability analysis, emphasizing the influence of the large wave number.
Compact Finite Differences
Fewer nodes used in computations, leading to higher efficiency.
Domain Decomposition
Suitable for multiscaled methods, such as micro- and macro- regions.
Mathematics Education
Analysing changes during the emergency transition to remote learning.
Mathematical Biology
Mathematical analysis of the Darwinian evolution of parasites, from simple to complex life cycle.