A smoothness indicator constant for sine functions

Abstract

Smoothness indicator is an essential part of weighted essentially non-oscillatory (WENO) scheme, whose target is to distinguish discontinuous profiles and continuous profiles. However, the magnitudes of most of the smoothness indicators will be decreased significantly when the stencil approaches critical points, which would be negative for the numerical simulation. The reason should be that the first derivative term which will drop to almost zero occupies a large proportion in these smoothness indicators. To decrease the variances on different stencils in smooth region, smoothness indicators could be required to be constant for a specific kind of smooth functions, and the best choice should be the sine functions. In the present paper, a new smoothness indicator on four-point stencil is constructed based on this criterion. Compared with the classical smoothness indicator (Jiang and Shu, 1996, [3]), the new one has a more succinct form, and takes less floating point operations which means that it will be more time efficient in the computation. Furthermore, the new smoothness indicator will get a larger variation as the sub-stencil moves from a smooth profile to a discontinuous profile, which would be helpful for stability near discontinuities. By using the new smoothness indicator, a seven-point WENO scheme is constructed which will reduce to the underlying linear scheme for monochromatic waves. As a result, it behaves the same as the underlying linear scheme for approximate dispersion relation (ADR). According to properties of the proposed smoothness indicator, this scheme should have excellent performance for profiles close to sine waves, better stability near discontinuities, and higher time efficiency. Numerical simulations predict the good performance of the proposed scheme.