Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws

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2009-419.pdf (344.67 KB)
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Shu, Chi-Wang
Xu, Zhiliang
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http://hdl.handle.net/1853/29403
Abstract
The hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM ’07] can effectively reduce spurious oscillations without local characteristic decomposition for numerical capturing of discontinuous solutions. However, there are still small re- maining overshoots/undershoots in the vicinity of discontinuities. HR with partial neighboring cells [Xu, Liu and Shu, JCP ’09] essentially overcomes this drawback for the third order case, and in the mean time further improves the resolution of the numer- ical solution. Extending the technique to higher order cases we observe the returning of overshoots/undershoots. In this paper, we introduce a new technique to work with HR on partial neighboring cells, which lowers the order of the remainder while maintaining the theoretical order of accuracy, essentially eliminates overshoots/undershoots for the fourth and fifth order cases (in one dimensional numerical examples) and reduces the numerical cost.
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NSF grant DMS-0810913; ARO grant W911NF-08-1-0520; NSF grant DMS-0809086; NSF grant DMS-0800612
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2009-07-31
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