Abstract
A serious and ubiquitous issue in existing mapped WENO schemes is that most of them can hardly preserve high resolutions, but in the meantime prevent spurious oscillations in the solving of hyperbolic conservation laws with long output times. Our goal for this article was to address this widely known problem. In our previous work, the order-preserving (OP) criterion was originally introduced and carefully used to devise a new mapped WENO scheme that performs satisfactorily in long simulations, and hence it was indicated that the OP criterion plays a critical role in the maintenance of low-dissipation and robustness for mapped WENO schemes. Thus, in our present work, we firstly defined the family of mapped WENO schemes, whose mappings meet the OP criterion, as OP-Mapped WENO. Next, we attentively took a closer look at the mappings of various existing mapped WENO schemes and devised a general formula for them. That helped us to extend the OP criterion to the design of improved mappings. Then, we created a generalized implementation of obtaining a group of OP-Mapped WENO schemes, named MOP-WENO-X, as they are developed from the existing mapped WENO-X schemes, where the notation “X” is used to identify the version of the existing mapped WENO scheme. Finally, extensive numerical experiments and comparisons with competing schemes were conducted to demonstrate the enhanced performances of the MOP-WENO-X schemes.
Highlights
The essentially non-oscillatory (ENO) schemes [1,2,3,4] and the weighted ENO (WENO)schemes [5,6,7,8] have been developed quite successfully in recent decades to solve the hyperbolic conservation problems, especially those that may generate discontinuities and smooth small-scale structures as time evolves in their solutions, even if the initial condition is smooth
It should be noted that we mainly provide the solutions of the fifth-order WENO methods (WENO5) in present study, the methodology proposed in this paper can be successfully extended to higher order WENO methods, such as WENO-7 or WENO-9, and because of the space limitations, we do not show their solutions here
For the sake of brevity though, we only present the solutions of the WENO-M, WENO-IM(2, 0.1), WENO-PPM5, WENOMAIM1 schemes and their associated MOP-WENO-X schemes in Figures 20 and 21, where the first rows give the final structures of the shock and vortex in density profile of the existing mapped WENO-X schemes, the second rows give those of the associated MOPWENO-X schemes, and the third rows give the cross-sectional slices of density plot along the plane y = 0.65 where x ∈ [0.70, 0.76]
Summary
The essentially non-oscillatory (ENO) schemes [1,2,3,4] and the weighted ENO (WENO)schemes [5,6,7,8] have been developed quite successfully in recent decades to solve the hyperbolic conservation problems, especially those that may generate discontinuities and smooth small-scale structures as time evolves in their solutions, even if the initial condition is smooth. The main purpose of this study was to find a general method to introduce the order-preserving (OP) mapping proposed in our previous work [9] for improving the existing mapped WENO schemes for the approximation of the hyperbolic conservation laws in the form ∂u ∂t + ∇ · F(u) = (1). There have been many works by Dumbser [10], Boscheri [11,12,13], Tsoutsanis [14,15], Titarev and Toro [16,17,18,19], Semplice [20,21], Puppo [22], Russo [23,24], and others on WENO approaches. Our main concern was to improve the performances of the (2r − 1)th-order mapped WENO schemes, so we briefly review recent developments in this field in the following
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