Improved sixth-order WENO scheme based on the numerical flux and smoothness indicators of modified stencils

In this paper, we propose a new sixth-order weighted essentially non-oscillatory (WENO) scheme for solving compressible aerodynamics. As with the previous six-order central upwind WENO scheme, this present scheme is a convex combination of four linear reconstructions. The stencil type is the same as WENO-Z6 scheme (Hu FX 2018), although the first three upwind stencils nominally use three cell values, the original secondary reconstruction has been modified to have fourth-order accuracy by adding a third-order correction term involving five cell values. In addition, corresponding smoothness indicators are provided, the new schemes are called WENO-MS-JS6 and WENO-MS-Z6. And by using the new reference smoothness indicator, the WENO-MS-Z6 scheme achieved the optimal convergence accuracy order in the smooth region including the first-order critical point (f(i)' = 0, f(i)'' not equal 0, f(i)''' not equal 0). Several numerical examples demonstrate that the new schemes have better resolution and the same robustness as the recently developed sixth order WENO schemes.