Improvement of the WENO-Z+ scheme. (English) Zbl 1521.76565
Summary: The WENO-Z+ scheme [F. Acker et al., J. Comput. Phys. 313, 726–753 (2016; Zbl 1349.65260)] was obtained by adding a new term into the WENO-Z weights. The added term has the role of raising the weights of less-smooth substencils. The WENO-Z+ scheme achieves superior results in the test problems that involve only a single wave component. For problems containing multiscale structures, however, the results of WENO-Z+ show little improvement compared with those of WENO-Z. In this short note, we investigate the reason why WENO-Z+ has little improvement in solving the multiscale problems and propose a set of modifications to it. A series of numerical tests confirm that the new scheme, which is named WENO-Z+I, has significantly improved multiscale resolution. The multiscale resolution of the fifth-order WENO-Z+I scheme is even significantly better than that of the seventh-order WENO-Z scheme. We also find that the WENO-Z+ type schemes can achieve better spectral properties than the corresponding linear upwind scheme.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Citations:
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