Hierarchical Reconstruction for Discontinuous Galerkin (DG) Method for Hyperbolic Conservation Laws and a New Formulation of DG Method
Zhiliang Xu
Department of Mathematics
Notre Dame University
In this talk, I will present some recent developments of hierarchical reconstruction (HR) [Liu etal., Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction. SIAM J. Numer. Anal., 45:2442-2467, 2007 and Xu etal.
Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells. J.C.P. 228:2194-2212, 2009] for limiting solutions computed by Runge-Kutta (RK) DG methods for hyperbolic conservation laws. The idea of HR is to decompose the task of a high degree polynomial reconstruction into a series of linear polynomial reconstruction process. The main features of HR are order preserving (for smooth solutions), compact, and without characteristic decomposition.
We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of these linear polynomials to preserve the order of accuracy on unstructured meshes. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.
In addition, I will discuss a new formulation of RKDG method based on the conservation constraint. With this new formation, we can achieve a larger CFL number for RKDG method of order >= 3.