Commun. Comput. Phys., 2 (2007), pp. 933-963.


Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

Yingjie Liu 1*, Chi-Wang Shu 2, Eitan Tadmor 3, Mengping Zhang 4

1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.
2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
3 Department of Mathematics, Institute for Physical Science and Technology, and Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, MD 20742, USA.
4 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.

Received 30 October 2006; Accepted (in revised version) 11 January 2007
Available online 20 March 2007

Abstract

This is the continuation of the paper ''Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction'' by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids. This takes a new finite volume approach for approximating non-smooth solutions. A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially non-oscillatory solutions even for $5$th-order schemes without characteristic decomposition.

AMS subject classifications: 65M06, 65M60

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Key words: Central scheme, discontinuous Galerkin method, ENO scheme, finite volume scheme, MUSCL scheme, TVD scheme.

*Corresponding author.
Email: yingjie@math.gatech.edu (Y. Liu), shu@dam.brown.edu (C.-W. Shu), tadmor@cscamm.umd.edu (E. Tadmor), mpzhang@ustc.edu.cn (M. Zhang)
 

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