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Center for Nonlinear Studies |
In this talk I will introduce the hierachical reconstruction (HR) for discontinuous Galerkin method (DG), central DG, central and finite volume schemes on regular or irregular meshes, in particular, the recent progress such as HR with partial neighboring cells for 3rd order non-staggered schemes, HR with quadratic remainder on partial neighboring cells for 4th and 5th order finite volume reconstructions. Some difficulties and open problems will be discussed when applying HR to various piecewise polynomial solutions of different orders (which e.g., are computed by a DG method or by a finite volume reconstruction method). HR is a general reconstruction procedure used as a limiter to remove spurious oscillations in the presence of shocks or other non-smooth solutions. Given a pecewise polynomial solution which may contain Gibbs phenomenon, it decomposes the job of limiting a high order polynomial hierarchically into smaller jobs each of them only involves the reconstruction of a linear polynomial, allowing the use of the MUSCL, second order ENO or other non-oscillatory "linear" reconstructions methods. HR does not use local characteristic decomposition for systems. Therefore it is compact and easy to implement for arbitrary meshes. Results discussed in the talk are based on a series of works by Yingjie Liu, Chi-Wang Shu, Eitan Tadmor, Zhiliang Xu and Mengping Zhang.
Host: Mikhail Shashkov