A New Mapped WENO Method for Hyperbolic Problems

In this study, a new family of rational mapping functions g(RM)(omega;k,m,s) is introduced for seventh-order WENO schemes. g(RM) is a more general family of mapping functions, which includes other mapping functions such as g(M) and g(IM) as special cases. The mapped WENO scheme WENO-IM(2,0.1), which uses g(IM), performs excellently at fifth order but rather poorly at seventh order. The reason for this loss of accuracy was found to be the over-amplification of very small weights by the mapping process, which can be traced back to the large slope of g(IM) at omega = 0. For m > 1, g(RM) can be designed to have a unit slope at omega = 0, which will preserve small weights with little to no amplification. It has been demonstrated through several one-dimensional linear advection test cases that the mapped WENO scheme WENO-RM(6,3,2 x 10(3)), which uses the mapping function g(RM)(omega;6,3,2 x 10(3)), outperforms both WENO-M and WENO-IM(2,0.1) at seventh order. The proposed scheme also performs better at a number of one-dimensional inviscid gas flow problems compared to other popular WENO schemes such as the WENO-Z scheme.

Automated summary

In this study, a new family of rational mapping functions g(RM)(omega;k,m,s) is introduced for seventh-order WENO schemes. It has been demonstrated that the mapped WENO scheme WENO-RM(6,3,2 x 10(3)), which uses the mapping function g(RM)(omega;6,3,2 x 10(3)), outperforms both WENO-M and WENO-IM(2,0.1) at seventh order. The proposed scheme also performs better at a number of one-dimensional inviscid gas flow problems compared to other popular WENO schemes such as the WENO-Z scheme.