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

TL;DR

The paper proposes the Fourier-Gegenbauer Integral-Galerkin (FGIG) method to solve the 1D advection-diffusion equation with periodic boundaries by marrying Fourier spatial expansion with Gegenbauer temporal integration, reformulated as a system of integral equations that eliminates time stepping. It establishes a robust mathematical framework, demonstrates exponential convergence for smooth solutions, and analyzes stability and conditioning, especially regarding Gegenbauer parameters. The semi-analytical FGIG (SA-FGIG) variant yields near-machine-precision solutions and provides a rigorous benchmark tool. Numerical experiments show superior accuracy and efficiency of FGIG compared with traditional time-stepping schemes, with SA-FGIG offering ultra-high accuracy at some computational cost. The work opens avenues for extensions to higher dimensions and diffusion-dominated problems, while acknowledging limitations in advection-dominated regimes and suggesting future enhancements such as adaptive meshes and hybrid approaches.

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 Péclet 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.

Figures (16)

• Figure 1: Flowchart of the FGIG Algorithm.
• Figure 2: The distribution of the eigenvalues (blue circles) and the extreme singular values (red x marks) of ${}_T\mathbf{Q}$ (first row) and ${}_T\mathbf{A}^{(1,M)}$ (second row) for $M = 4$ and $\lambda = -0.4, -0.2, 0:0.5:2$. All plots were generated using $L = 2$, $T = 0.2, \mu = 0, \nu = 1$, and $N = 4$.
• Figure 3: The distribution of the eigenvalues (blue circles) and the extreme singular values (red x marks) of ${}_T\mathbf{Q}$ (first row) and ${}_T\mathbf{A}^{(1,M)}$ (second row) for $M = 40$ and $\lambda = -0.4, -0.2, 0:0.5:2$. All plots were generated using $L = 2$, $T = 0.2, \mu = 0, \nu = 1$, and $N = 4$.
• Figure 4: The distribution of the eigenvalues (blue circles) and the extreme singular values (red x marks) of ${}_T\mathbf{Q}$ (first row) and ${}_T\mathbf{A}^{(1,M)}$ (second row) for $M = 80$ and $\lambda = -0.4, -0.2, 0:0.5:2$. All plots were generated using $L = 2$, $T = 0.2, \mu = 0, \nu = 1$, and $N = 4$.
• Figure 5: The values of the smallest singular values of $\mathbf{Q}$ for $\lambda = -0.49,-0.499,-0.4999$ and $M = 4, 40$, and $80$.

Theorems & Definitions (16)

• Theorem 5.1
• proof
• Theorem 6.1: elgindy2024numerical
• Theorem 6.2: Elgindy2023a
• Corollary 6.1: elgindy2024numerical
• Corollary 6.2: Elgindy2023a
• Theorem 6.3: Elgindy20161
• Theorem 6.4: Elgindy2023a
• Theorem A.1
• proof