Abstract
Weighted essentially non-oscillatory (WENO) schemes are a class of numerical schemes designed for solving problems with discontinuities. The spectral properties of them are hard to analyze because of their nonlinearities. A method of analyzing the average scheme in effect is proposed, by which we found that WENO schemes with a special kind of smoothness indicators can even obtain better spectral properties than their underlying linear schemes. This facilitates the design of WENO schemes with good spectral properties. The formulas of these smoothness indicators on sub-stencils have the same form except for different subscripts, which makes the nonlinear weights slightly close to the mean value. With this weight distribution, the resultant schemes behave like optimized upwind schemes with low dissipation and optimized dispersion properties. However, these WENO schemes may suffer from anti-dissipation and accuracy order degradation. Hence, they are suggested to be combined with the mapped WENO or the Z-type WENO methodology to address these problems. Some WENO schemes are taken as examples to illustrate the proposed methodology, and the results are in good agreement with those from the approximated dispersion relation (ADR) method.
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