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 le...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 overamplify 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.展开更多
文摘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 overamplify 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.
