Motivated by the moment limiter of Biswas, Devine and Flaherty
[Appl. Numer. Math. 14(1994)], we develop a general non-oscillatory
hierarchical reconstruction (HR) procedure for removing the spurious
oscillations in the high degree polynomial of a cell computed by the
central DG scheme. This procedure is fully multidimensional and can
be applied to any shapes of cells at least in theory. HR uses the
most compact stencil and thus fitts very well with DG. Further more,
it doesn't need any charcteristic decomposition even for very high
order, such as 5th order, even though there could be small overshoots/undershoots for very high order when there are interactions of
discontinuities. The basic idea and major operations are to decompose a
high degree polynomial reconstruction into hierarchical linear reconstructions,
such as the MUSCL reconstruction.
HR can be applied to central and finite volume schemes
as well resulting in a new finite volume approach. These finite volume
schemes can be used for unstructured meshes with essentially no restriction
on meshes, and can be done without charcteristic decomposition. We will
demonstrate through numerical experiments the effectiveness of the HR.
If time allows, I can also disuss a recently developed BFECC algorithm
and its application to fluid and interface simulations for computer
graphics.
Joint with Chi-Wang Shu, Eitan Tadmor and Mengping Zhang
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