A hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations

TL;DR

This paper introduces a hybridized discontinuous Galerkin method tailored for 2D fractional convection-diffusion equations, demonstrating stability and convergence through theoretical analysis and numerical experiments on triangular meshes.

Contribution

It develops a novel hybridized discontinuous Galerkin scheme for 2D fractional PDEs, combining characteristic methods with stability proofs and convergence analysis.

Findings

Proves scheme stability and establishes convergence order of O(h^{k+1/2} + Δt)

Numerical experiments confirm theoretical convergence rates

Demonstrates advantages of the method on triangular meshes

Abstract

A hybridized discontinuous Galerkin method is proposed for solving 2D fractional convection-diffusion equations containing derivatives of fractional order in space on a finite domain. The Riemann-Liouville derivative is used for the spatial derivative. Combining the characteristic method and the hybridized discontinuous Galerkin method, the symmetric variational formulation is constructed. The stability of the presented scheme is proved. Theoretically, the order of $\mathcal{O}(h^{k+1/2}+\Delta t)$ is established for the corresponding models and numerically the better convergence rates are detected by carefully choosing the numerical fluxes. Extensive numerical experiments are performed to illustrate the performance of the proposed schemes. The first numerical example is to display the convergence orders, while the second one justifies the benefits of the schemes. Both are tested with triangular meshes.