A New Hybrid WENO Scheme on a Four-Point Stencil for Euler Equations

Abstract

To enhance the performance of the third-order weighted essentially non-oscillatory (WENO) scheme on the four-point stencil, a new hybrid central-type scheme is proposed in this paper. Developments are conducted in the following two aspects. First, a new fourth-order central WENO scheme, named as WENO4-LC, is devised for shock capturing, and a low dissipative third-order grid-centered upwind scheme is adopted for the smooth region. Second, a discontinuity indicator is extended to the four-point stencil version, which is simpler than its counterpart in our former article (Guo et al. in J Sci Comput 83, 28, 2020), for switching between the linear and nonlinear branch of the hybrid scheme. Numerical tests of the Euler and Navier–Stokes benchmarks show that the new indicator is good on detecting the discontinuities. Moreover, the performances of the central-type WENO4-LC and its corresponding hybrid scheme have obvious improvements compared with those of WENO3-JS and WENO3-L, and even behave slightly better than classical WENO5-JS on resolving the fluid structures. Meanwhile, the hybrid scheme is efficient which only costs 76% CPU time of WENO3-JS and about 58% of WENO5-JS.