In 18%DIFDELCMD < ]%%% , 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 schemeis 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 .
 %DIFDELCMD < ]%%% A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions and in the meantime prevent spurious oscillations on solving hyperbolic conservation laws with long output times. Our goal in this article is to address this widely concerned problem 3,4 , 15,29,16,18 .We firstly take a closer look at the mappings of various existing mapped WENO schemes and devise a general formula for them. It helps us to extend the order-preserving (OP) criterion, originally defined and carefully examined in 18], into the design of the mappings.Next, we propose the implementation of obtaining the new mappings satisfying the OP criterion from those of the existing mapped WENO-X schemes where the notation "X" is used to identify the version of the existing mapped WENO scheme, e.g., X = M 11], PM6 3], or PPM5 15], et al. Then we build the resultant mapped WENO schemes and denote them as MOP-WENO-X. The numerical solutions of the one-dimensional linear advection equation with different initial conditions and some standard numerical experiments of two-dimensional Euler system, computed by the MOP-WENO-X schemes , are compared with the ones generated by their corresponding WENO-X schemes and the WENO-JS scheme. To summarize, the MOP-WENO-X schemes gain definite advatages in terms of attaining high resolutions and meanwhile avoiding spurious oscillations near discontinuities for long output time simulations of the one-dimensional linear advection problems, as well as significantly reducing the post-shock oscillations in the simulations of the two-dimensional steady problems with strong shock waves .
D
In
D
18
R
, 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
A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve
R
avoid spurious oscillations near discontinuities when solving the one-dimensional linear advection problems, expecially for
in the meantime prevent spurious oscillations on solving hyperbolic conservation laws with
R
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
Our goal in this article is to address this widely concerned problem
A
3,4
R
we
15,29,16,18
A
.We firstly
A
the mappings of
R
in references to obtain
and devise
R
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 schemeis named
them. It helps us to extend the order-preserving (OP) criterion, originally defined and carefully examined in
A
18
A
, into the design of the mappings.Next, we propose the implementation of obtaining the new mappings satisfying the OP criterion from those of the existing mapped WENO-X schemes where the notation "X" is used to identify the version of the existing mapped WENO scheme, e.g., X = M
A
11
A
, PM6
A
3
A
, or PPM5
A
15
A
, et al. Then we build the resultant mapped WENO schemes and denote them as
R
Numerical
The numerical
A
and some standard numerical experiments of two-dimensional Euler system,
A
,
R
the
their
D
Some standard numerical experiments of two-dimensional Euler system such as the shock-vortex interaction and the 2D Riemann problems are presented.
D
also enjoy the advantages of the MOP-WENO-ACMk scheme proposed in
R
18
gain definite advatages in terms of attaining high resolutions and meanwhile avoiding spurious oscillations near discontinuities for long output time simulations of the one-dimensional linear advection problems, as well as significantly reducing the post-shock oscillations in the simulations of the two-dimensional steady problems with strong shock waves