Locally Order-Preserving Mapping for WENO Methods

Most of the existing mapped WENO schemes suffer from either losing high resolutions or generating spurious oscillations in long-run simulations of hyperbolic problems. The purpose of this paper is to amend this commonly reported issue. We firstly present the definition of 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 where "X" stands for the version of the existing mapped WENO scheme. 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 PoAOP-WENO-X. The numerical results of the 1D linear advection problem with different initial conditions and some standard 2D problems modeled via Euler equations, calculated by the PoAOP-WENO-X schemes, are compared with the ones generated by their non-OP counterparts and the WENO-JS scheme. In summary, the PoAOP-WENO-X schemes enjoy great advantages in terms of 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.