Discontinuity-Detecting Method for a Four-Point Stencil and Its Application to Develop a Third-Order Hybrid-WENO Scheme

First this paper analyzes the reason for the accuracy losing of the third-order weighted essentially non-oscillatory (WENO) scheme. It is shown that one reason is that the local smoothness indicators of the third-order WENO scheme cannot correctly treat the smooth three-point stencil containing a non-nodal critical point, here, 'non-nodal' means the critical point is not a grid point. And then a discontinuity-detecting method for a four-point stencil is proposed and applied to develop the high order accurate hybrid-WENO scheme by combining the third-order WENO scheme and a third-order upstream scheme. This four-point stencil is actually the stencil used for constructing the third-order WENO scheme (positive and negative numerical fluxes), hence the resulting hybrid-WENO scheme proposed by this paper does not introduce new grid point. Numerical examples show that the detecting method and the hybrid scheme are robust for problems with shocks, and the hybrid scheme obtains real third-order convergence rate for smooth solutions containing critical points