Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy

In this paper we design a class of numerical schemes that are higher-order extensions of the weighted essentially non-oscillatory (WENO) schemes of G. -S. Jiang and C.-W, Shu (1996) and X.-D. Liu, S. Osher, and T. Chan (1994), Used by themselves, the schemes may not always be monotonicity preserving but coupled with the monotonicity preserving bounds of A. Suresh and H. T. Huynh (1997) they perform very well, The resulting monotonicity preserving weighted essentially non-oscillatory (MPWENO) schemes have high phase accur acy and high order of accuracy. The higher-order members of this family are almost spectrally accurate for smooth problems, Nevertheless, they, have r obust shock capturing ability. The schemes are stable under normal CFL numb ers. They are also efficient and do not have a computational complexity tha t is substantially greater than that of the lower-order members of this sam e family of schemes. The higher accuracy that these schemes offer coupled w ith their relatively low computational complexity makes them viable competi tors to lower-order schemes, such as the older total variation diminishing schemes, for problems containing both discontinuities and rich smooth regio n structure. We describe the MPWENO schemes here as well as show their abil ity to reach their designed accuracies for smooth flow. We also examine the role of steepening algorithms such as the artificial compression method in the design of very high order schemes. Several test problems in one and tw o dimensions are presented. For multidimensional problems where the flow is not aligned with any of the grid directions it is shown that the present s chemes have a substantial advantage over lower-order schemes. It is argued that the methods designed here have great utility for direct numerical simu lations and large eddy simulations of compressible turbulence. The methodol ogy developed hen is applicable to other hyperbolic systems, which is demon strated by showing that the MPWENO schemes also work very well on magnetohy drodynamical test problems. (C) 2000 Academic Press.