310 research outputs found
A hybrid Hermite WENO scheme for hyperbolic conservation laws
In this paper, we propose a hybrid finite volume Hermite weighted essentially
non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic
conservation laws. The zeroth-order and the first-order moments are used in the
spatial reconstruction, with total variation diminishing Runge-Kutta time
discretization. The main idea of the hybrid HWENO scheme is that we first use a
shock-detection technique to identify the troubled cell, then, if the cell is
identified as a troubled cell, we would modify the first order moment in the
troubled cell and employ HWENO reconstruction in spatial discretization;
otherwise, we directly use high order linear reconstruction. Unlike other HWENO
schemes, we borrow the thought of limiter for discontinuous Galerkin (DG)
method to control the spurious oscillations, after this procedure, the scheme
would avoid the oscillations by using HWENO reconstruction nearby
discontinuities and have higher efficiency for using linear approximation
straightforwardly in the smooth regions. In addition, the hybrid HWENO scheme
still keeps the compactness. A collection of benchmark numerical tests for one
and two dimensional cases are performed to demonstrate the numerical accuracy,
high resolution and robustness of the proposed scheme.Comment: 38 page
Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes
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 nite volume schemes on non-staggered grids. This takes a new nite volume approach for approximating non-smooth solutions. A critical step for high order nite volume schemes is to reconstruct a nonoscillatory
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 5th order schemes without
characteristic decomposition.The research of Y. Liu was supported in part by NSF grant DMS-0511815. The research of C.-W. Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science
and Technology of China (grant 2004-1-8) and the Institute of Computational Mathematics and Scienti c/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345. The research of E. Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. The research of M. Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8
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