A new adaptively central-upwind sixth-order WENO scheme

In this paper, we propose a new sixth-order WENO scheme for solving one dimensional hyperbolic conservation laws. The new WENO reconstruction has three properties: (1) it is central in smooth region for low dissipation, and is upwind near discontinuities for numerical stability; (2) it is a convex combination of four linear reconstructions, in which one linear reconstruction is sixth order, and the others are third order; (3) its linear weights can be any positive numbers with requirement that their sum equals one. Furthermore, we propose a simple smoothness indicator for the sixth-order linear reconstruction, this smooth indicator not only can distinguish the smooth region and discontinuities exactly, but also can reduce the computational cost, thus it is more efficient than the classical one.