Preprint 2004-043

Central Schemes on Overlapping Cells

Abstract: Nessyahu and Tadmor's central scheme (J. Comput. Phys, 87(1990)) has the benefit of not using Riemann solvers or characteristic decomposition for solving hyperbolic conservation laws and related convection diffusion equations. But the staggered averaging causes large dissipation when the time step size is small comparing to the mesh size. The recent work of Kurganov and Tadmor (J. Comput. Phys, 160(2000)) overcomes the problem by use of a variable control volume and obtains a semi-discrete non-staggered central scheme. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the $O(1/\Delta t)$ dependence of the dissipation. Semi-discrete form of the central scheme can also be obtained to which the TVD Runge-Kutta time discretization of Shu and Osher (J. Comput. Phys, 77(1988)) can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh, thus could also be useful for unstructured mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally more compact reconstruction can be achieved. We demonstrate through numerical examples that combining two classes of the overlapping cells in the reconstruction can achieve higher resolution.

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Author(s):

Yingjie Liu, <yingjie@math.gatech.edu>

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<yingjie@math.gatech.edu> September 6 2004.