Yingjie Liu et al 2009 Nonlinearity 22 2799 doi:10.1088/0951-7715/22/12/001
Yingjie Liu1, Chi-Wang Shu2 and Zhiliang Xu3
Show affiliationsRecommended by A L Bertozzi
The hierarchical reconstruction (HR) (Liu et al 2007 SIAM J. Numer. Anal. 45 2442–67) can effectively reduce spurious oscillations without local characteristic decomposition for numerical capturing of discontinuous solutions. However, there are still small remaining overshoots/undershoots in the vicinity of discontinuities. HR with partial neighbouring cells (Xu et al 2009 J. Comput. Phys. 228 2194–212) essentially overcomes this drawback for the third order case, and in the mean time further improves the resolution of the numerical 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 neighbouring cells, which lowers the order of the remainder of the polynomial in the current cell while maintaining the theoretical order of accuracy, essentially eliminates overshoots/undershoots for the fourth and fifth order cases and reduces the numerical cost.
41A10 Approximation by polynomials (For approximation by trigonometric polynomials, see 42A10)
41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Issue 12 (December 2009)
Received 31 July 2009, in final form 21 September 2009
Published 30 October 2009
Yingjie Liu et al 2009 Nonlinearity 22 2799
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