An Improved WENO-Z+ Scheme

The WENO-Z+ scheme [F. Acker, R. B. de R. Borges, and B. Costa, An improved WENO-Z scheme, J. Comput. Phys., 313 (2016), pp. 726–753] with two different versions further raised the nonlinear weights with respect to the nonsmooth or less-smooth substencils, by introducing an additional term into the weights formula of the well-validated WENO-Z scheme. These two WENO-Z+ schemes both produce less dissipative solutions than WENO-JS and WENO-Z. However, the recommended one which achieves superior resolutions in the high-frequency-wave regions fails to recover the designed order of accuracy where there exists a critical point, while the other one which obtains the designed order of accuracy at or near critical points is unstable near discontinuities. In the present study, we find that the WENO-Z+ schemes over-amplify the contributions from less-smooth substencils through their additional terms, and hence their improvements of both stability and resolution have been greatly hindered. Then, we develop improved WENO-Z+ schemes by making a set of modifications to the additional terms to avoid the over-amplification of the contributions from less-smooth substencils. The proposed schemes, denoted as WENO-IZ+, maintain the same convergence properties as the corresponding WENO-Z+ schemes. Numerical examples confirm that the new schemes are much more stable near discontinuities and far less dissipative in the region with high-frequency waves than the WENO-Z+ schemes. In addition, improved results have been obtained for one-dimensional linear advection problems, especially over long output times. The excellent performance of the new schemes is also demonstrated in the simulations of 1D and 2D Euler equation test cases.