Research Projects

This proposal aims to develop high-order multiscale asymptotic-preserving numerical methods for stiff reaction-diffusion systems arising in both fluid dynamics and financial markets. Many real-world models exhibit multiscale behaviour, where disparate temporal and spatial scales introduce stiffness, posing significant challenges for standard numerical methods. This is particularly evident in low-Mach-number Navier-Stokes flows and financial markets under extreme volatility and liquidity constraints. The core motivation is to develop structure and asymptotic preserving methods that remain accurate and stable in both singular perturbation regimes and full dynamic models. In the context of fluid dynamics, we consider compressible and incompressible Navier-Stokes equations coupled with reaction-diffusion phenomena such as combustion or species transport. In financial mathematics, the focus is on option pricing influenced by liquidity shocks and transaction costs, which introduce similar multiscale and stiff behaviour. The proposed framework unifies tools from asymptotic analysis, implicit-explicit (IMEX) time integration, and compact finite difference schemes to construct methods that preserve energy, positivity, and conservation laws. The proposed schemes treat stiff components implicitly and non-stiff components explicitly without requiring nonlinear solvers and matrix inversions. This enables stable, large-time simulations while maintaining computational efficiency. Applications include low-speed aerodynamics, ocean-climate models, jump-diffusion pricing models, and porous media flows. The project is expected to yield robust numerical tools, rigorously validated on benchmark problems, with accompanying theoretical insights into convergence and stability across asymptotic regimes. It will also provide theoretical insights into the convergence and stability of these schemes across various limiting regimes, contributing to the advancement of computational methodologies and academic publications.