Central Discontinuous Galerkin Methods on Overlapping Cells with a Non-Oscillatory Hierarchical Reconstruction
MARYLAND UNIV COLLEGE PARK MD CENTER FOR SCIENTIFIC COMPUTATIONAL AND MATHEMATICAL MODELING
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The central scheme of Nessyahu and Tadmor J. Comput. Phys, 87 1990 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 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. 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. Comput. Phys. 141 1998 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-processes 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, 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.
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