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A modified fifth‐order WENO‐Z scheme is proposed by analogy with the non‐normalized weights of the reformulated fifth‐order adaptive order (AO) WENO scheme. We show that if the original fifth‐order WENO‐AO scheme is rewritten as the form of the conventional WENO combination, the resulting non‐normalized weights can be divided into three parts: a constant one term, a local stencil smoothness measure term and a global stencil smoothness measure term. In order to make use of the latter two terms for constructing a modified WENO‐Z scheme with enhanced performance, we change the form of the third term and introduce an adaptive scaling factor to adjust the contributions from the second and third terms. Numerical examples show that the modified fifth‐order WENO‐Z scheme has the advantage of high resolution in smooth regions and sharp capturing of discontinuities, and it can obtain evidently better results for shocked flows with small‐scale structures compared with the recently developed WENO‐Z+ and WENO‐Z+M schemes. [ABSTRACT FROM AUTHOR]

This paper presents a new hybrid scheme which combines the fifth-order Targeted Essentially Non-Oscillatory (TENO) reconstruction and the tangent of hyperbola for interface capturing (THINC) reconstruction by using a modified boundary variation diminishing (MBVD) algorithm to select one of the two candidate reconstructions. The THINC scheme has a jump-like distribution can approximate a discontinuity well within a mesh cell with high fidelity. The original BVD algorithm is based on minimisation of jumps of reconstructed variables at cell interfaces in order to reduce the dissipation in numerical flux. The present MBVD algorithm minimises the dissipation terms of the HLL flux which includes jumps of both variables and fluxes at cell interfaces. A number of benchmark examples demonstrate that the MBVD algorithm is more robust than the original BVD algorithm. Applications in stiff detonation problems show that the present hybrid scheme can capture detonation fronts better than the single TENO scheme. [ABSTRACT FROM AUTHOR]

Summary: The classical third‐order weighted essentially nonoscillatory (WENO) scheme is notoriously dissipative as it loses the optimal order of accuracy at critical points and its two‐point finite difference in the smoothness indicators is unable to differentiate the critical point from the discontinuity. In recent years, modifications to the smoothness indicators and weights of the classical third‐order WENO scheme have been reported to reduce numerical dissipation. This article presents a new reference smoothness indicator for constructing a low‐dissipation third‐order WENO scheme. The new reference smoothness indicator is a nonlinear combination of the local and global stencil smoothness indicators. The resulting WENO‐Rp3 scheme with the power parameter p=1.5 achieves third‐order accuracy in smooth regions including critical points and has low dissipation, but numerical results show this scheme cannot keep the ENO property near discontinuities. The recommended WENO‐R3 scheme (p=1) keeps the ENO property and performs better than several recently developed third‐order WENO schemes. [ABSTRACT FROM AUTHOR]