Central Discontinuous Galerkin Methods on Overlapping Cells with a Non-Oscillatory Hierarchical Reconstruction
Yingjie Liu, Chi-Wang Shu, Eitan Tadmor, and Mengping Zhang
Abstract: The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990); doi:10.1016/0021-9991(90)90260-8] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000); doi:10.1006/jcph.2000.6459] employs a variable control volume, which in turn yields a semi-discrete non-staggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005); doi:10.1016/j.jcp.2005.03.014]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comp. Phys. 141 (1998); doi:10.1006/jcph.1998.5892] and the references therein.
In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which post-process the central DG solution to remove spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14 (1994); doi:10.1016/0168-9274(94)90029-9], but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multi-layer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes and retains the overall pre-processed order of accuracy while effectively removes spurious oscillations around shocks.