Research Projects

Spectral methods have emerged as a powerful class of numerical techniques for solving partial differential equations (PDEs), especially in the simulation of complex fluid dynamics systems. Their high-order accuracy, rapid convergence for smooth solutions, and efficient representation of global behavior make them particularly appealing for modeling incompressible flows, turbulence, and wave propagation governed by nonlinear PDEs. This project, titled “Hybrid Spectral Methods for Coupled Nonlinear PDEs,” focuses on developing a robust numerical solver for complex systems governed by coupled partial differential equations, specifically the Navier-Stokes equations coupled with reaction-diffusion models. These systems arise in various fields, including fluid dynamics, chemical engineering, and environmental science, where chemical reactions influence fluid viscosity and flow stability. The core objective is to design a hybrid spectral solver that combines the high accuracy of pseudospectral methods with the flexibility of local discretization techniques to address the challenges posed by nonlinearities, sharp gradients, and complex boundary conditions. The approach builds upon the use of Fourier and Chebyshev polynomial expansions to represent unknown functions efficiently in space, offering rapid convergence for smooth problems. To handle non-smooth data and discontinuities (e.g., those causing the Gibbs phenomenon), the solver will incorporate mollification strategies and adaptive filtering. The mathematical formulation includes a coupled incompressible Navier-Stokes and reaction-diffusion-convection system, where the viscosity is modeled as a function of the chemical species concentration. This setting captures phenomena such as the Kelvin–Helmholtz instability (KHI) in layered flows, where local changes in viscosity due to chemical reactions can lead to flow instabilities and complex pattern formation.