ACMS Colloquium: Hybrid Weighted Essentially Non-Oscillatory Schemes with Different Indicators


Location: 127 Hayes-Healy Center


Jianxian Qiu, Professor of Mathematics at the Xiamen University, will give a colloquium titled, "Hybrid Weighted Essentially Non-Oscillatory Schemes with Different Indicators" at 4:00 PM in 127 Hayes-Healy Center.


A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation.  The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO.  For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillatory.  But the cost of computation of nonlinear weights and local characteristic decompositions is very expensive.

In the presentation, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and nonlinear weights for the part of the procedure, hence reducing the cost bust still maintaining the non-oscillatory properties for the problem with strong shocks. The idea is to identify discontinuity by a discontinuity indicator, then reconstruct numerical flux by WENO approximation at discontinuity and up-wind linear approximation at smoothness. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu {SIAM J Sci. Comput. 27 (2005) 995-1013}. The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, which an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost.  Detail numerical studies in one and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property.
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