{
  "id": "2501.02307",
  "version": "v1",
  "published": "2025-01-04T15:05:15.000Z",
  "updated": "2025-01-04T15:05:15.000Z",
  "title": "Fourier-Gegenbauer Integral-Galerkin Method for Solving the Advection-Diffusion Equation With Periodic Boundary Conditions",
  "authors": [
    "Kareem T. Elgindy"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-04T15:05:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periodic boundary conditions",
      "fourier-gegenbauer integral-galerkin method",
      "one-dimensional advection-diffusion equation",
      "relatively low peclet numbers",
      "ensure high accuracy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}