On the positivity of linear weights in WENO approximations |
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Authors: | Yuan-yuan Liu Chi-wang Shu Meng-ping Zhang |
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Institution: | [1]Department of Mathematics, University of Science and Technology of China, Hefei 230026, China [2]Division of Applied Mathematics, Brown University, Providence, RI 02912, USA |
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Abstract: | High order accurate weighted essentially non-oscillatory (WENO) schemes have been used extensively in numerical solutions
of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be
applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review
the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in
a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second
derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The
results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation
to first and second derivatives, and integration procedures.
Supported by the National Natural Science Foundation of China (No. 10671190) and Natural Science Foundation grant DMS-0809086
and ARO grant W911NF-08-1-0520. |
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Keywords: | Weighted essentially non-oscillatory (WENO) scheme hyperbolic partial differential equations WENO interpolation WENO reconstruction WENO approximation to derivatives WENO integration |
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