A hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations
Provides a stable numerical scheme for fractional PDEs, but is an incremental extension of existing methods to a specific equation class.
The paper proposes a hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations, achieving theoretical convergence of O(h^{k+1/2}+Δt) and better numerical rates with careful flux choices.
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 $\mathcal{O}(h^{k+1/2}+Δt)$ 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.