An efficient smoothness indicator mapped WENO scheme for hyperbolic conservation laws

Abstract

The mapping function method is a common approach to improve the accuracy of WENO type schemes. However, with the demand of the accuracy improvement higher, the mapping function becomes more and more complex, and the calculation cost also increases significantly. In this study, a novel smoothness indicator mapping method called WENO-ISM is proposed based on the WENO scheme. It employs a simple function which amplifies the original smoothness to improve the scheme's accuracy. In addition to the difference of the mapping function, the superior innovation of the designed scheme is that the mapping object is changed to the smoothness indicator instead of the nonlinear weight calculated from the WENO-JS scheme. The underlying idea of this method is to use this mapping function to directly regard the optimal weight of the stencil in the relatively smooth region as the nonlinear weight. For these, the calculation cost can be greatly reduced without loss of accuracy. In the meanwhile, the simple mapping function itself is with higher accuracy and lower cost. Numerical experiments with one-dimensional linear advection and ADR analysis show that this scheme is superior to other mapped WENO schemes in accuracy and especially in computation cost. Furthermore, improved results of this scheme are obtained by several typical one-dimensional and two-dimensional numerical cases.