A general method to introduce order-preserving mapping for improving the mapped WENO schemes

In [18], the authors have designed a new finite-volume mapped WENO scheme by introducing a new criterion which is defined as order-preserving (OP) in the design of the mapping. With this new criterion, a new mapping function was devised and the relevant mapped WENO scheme enjoyed the advantage that has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities when solving the one-dimensional linear advection problems, expecially for long output times. Moreover, this new scheme, dubbed as MOP-WENO-ACMk, can significantly reduce the post-shock oscillations in the simulations of the two-dimensional steady problems with strong shock waves. In this article, we take a closer look at various existing mapped WENO schemes in references to obtain a general formula for their mapping functions that allows the extension of the OP criterion to all existing mapped WENO schemes. The improved mapped WENO scheme considerring the OP criterion based on the existing mapped WENO-X scheme is named MOP-WENO-X. Numerical solutions of the one-dimensional linear advection equation with different initial conditions computed by the MOP-WENO-X schemes are compared with the ones generated by the corresponding WENO-X schemes and the WENO-JS scheme. Some standard numerical experiments of two-dimensional Euler system such as the shock-vortex interaction and the 2D Riemann problems are presented. To summarize, the MOP-WENO-X schemes also enjoy the advantages of the MOP-WENO-ACMk scheme proposed in [18].