Title:
Central Schemes on Overlapping Cells
Central Schemes on Overlapping Cells
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Liu, Yingjie
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Abstract
Nessyahu and Tadmor's central scheme (J. Comput. Phys, 87(1990)) has the benefi t 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/∆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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Date Issued
2004
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