Reconstructed Discontinuous Galerkin Methods Based on First-Order Hyperbolic System for Advection-Diffusion Equations

In this study, reconstructed Discontinuous Galerkin (rDG) methods are developed for solving advection-diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The developed hyperbolic rDG methods are reliable, accurate, efficient, and robust by combining the advantages of both FOHS and rDG methods. The presented methods have the same number of degrees-of-freedom as the conventional DG methods. Both hybrid least-squares reconstruction and variational reconstruction has been implemented in the study to deliver high order numerical solution while keeping the total degrees of freedom relatively small. A number of advection-diffusion test cases with a wide range of Reynolds numbers, including boundary layer type problems are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the advection-diffusion equations.

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