Two-step weighting method for constructing fourth-order hybrid central WENO scheme

Abstract

The hybridization of the fourth-order central scheme and a third-order WENO (weighted essentially non-oscillatory) scheme aims to obtain fourth-order accuracy in smooth regions and keep ENO property near shock waves. However, according to our numerical tests, such existing hybrid schemes may generate spurious waves near shocks. To overcome this problem, in this paper, a new two-step method is proposed for constructing a fourth-order hybrid central WENO scheme. The first step is using the traditional weighting method to get the third-order WENO scheme. The second step is to get a hybrid scheme by weighting the third-order upwind scheme and the fourth-order central scheme. Then, the final hybrid WENO scheme is obtained by combining the two weighted schemes through a discontinuity-detecting method for a four-point stencil. Various one- and two-dimensional numerical tests are presented to show that the new scheme can avoid the generation of spurious waves near shock waves, keep the ENO property, and get almost fourth-order accuracy in smooth regions including critical points.