Fourier-Gegenbauer Integral-Galerkin Method for Solving the Advection-Diffusion Equation With Periodic Boundary Conditions

This study presents the Fourier-Gegenbauer Integral-Galerkin (FGIG) method, a novel and efficient numerical framework for solving the one-dimensional advection-diffusion equation with periodic boundary conditions. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Distinctively, this approach eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of shifted-Gegenbauer-Gauss quadrature to ensure high accuracy and stability for relatively low P\'eclet numbers. Numerical experiments validate the method's superior performance over traditional techniques, demonstrating its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering.