Locally order-preserving mapping for WENO methods

Abstract

In our previous studies (Li and Zhong, 2021a; Li and Zhong, 2021b), the commonly reported issue that most of the existing mapped WENO schemes suffer from either losing high resolutions or generating spurious oscillations in long-run simulations of hyperbolic problems has been successfully addressed, by devising the improved mapped WENO schemes, namely MOP-WENO-X, where “X” stands for the version of the existing mapped WENO scheme. However, all the MOP-WENO-X schemes bring about the serious deficiency that their resolutions in the region with smoothly varying high-frequency waves are dramatically decreased compared to their associated WENO-X schemes. The purpose of this paper is to overcome this drawback. We first present the locally order-preserving (LOP) mapping. Then, by using a new proposed posteriori adaptive technique, we apply this LOP property to obtain the new mappings from those of the WENO-X schemes. The essential idea of the posteriori adaptive technique is to identify the global stencil in which the existing mappings fail to preserve the LOP property, and then replace the mapped weights with the weights of the classic WENO-JS scheme to recover the LOP property. We build the resultant mapped WENO schemes and denote them as LOP-WENO-X. The numerical experiments demonstrate that the resolutions in the region with high-frequency but smooth waves of the LOP-WENO-X schemes are similar to or even better than those of their associated WENO-X schemes and naturally much higher than the MOP-WENO-X schemes. Furthermore, the LOP-WENO-X schemes gain all the great advantages of the MOP-WENO-X schemes, such as attaining high resolutions and in the meantime preventing spurious oscillations near discontinuities when solving the one-dimensional linear advection problems with long output times, and significantly reducing the post-shock oscillations in the simulations of the two-dimensional problems with shock waves.