Generalized implementation of the order-preserving mapping for mapped WENO schemes
share
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.
READ FULL TEXT