Fengxiang Zhao

Fengxiang Zhao, Liang Pan, Shuanghu Wang

In this paper, a third-order weighted essentially non-oscillatory (WENO) scheme is developed for hyperbolic conservation laws on unstructured quadrilateral and triangular meshes. As a starting point, a general stencil is selected for the cell with any local topology, and a unified linear scheme can be constructed. However, in the traditional WENO scheme on unstructured meshes, the very large and negative weights may appear for the mesh with lower quality, and the very large weights make the WENO scheme unstable even for the smooth tests. In the current scheme, an optimization approach is given to deal with the very large linear weights, and the splitting technique is considered to deal with the negative weights obtained by the optimization approach. The non-linear weight with a new smooth indicator is proposed as well, in which the local mesh quality and discontinuities of solutions are taken into account simultaneously. Numerical tests are presented to validate the current scheme. The expected convergence rate of accuracy is obtained, and the absolute value of error is not affected by mesh quality. The numerical tests with strong discontinuities validate the robustness of current WENO scheme.

Fengxiang Zhao, Liang Pan, Zheng Li et al.

In this paper, A new sixth-order weighted essentially non-oscillatory (WENO) scheme, refered as the WENO-6, is proposed in the finite volume framework for the hyperbolic conservation laws. Instead of selecting one stencil for each cell in the classical WENO scheme [10], two independent stencils are used for two ends of the considering cell in the current approach. Meanwhile, the stencils, which are used for the reconstruction of variables at both sides of interface, are symmetrical. Compared with the classical WENO scheme [10], the current WENO scheme achieves one order of improvement in accuracy with the same stencil. The reconstruction procedure is defined by a convex combination of reconstructed values at cell interface, which are constructed from two quadratic and two cubic polynomials. The essentially non-oscillatory property is achieved by the similar weighting methodology as the classical WENO scheme. A variety of numerical examples are presented to validate the accuracy and robustness of the current scheme.

Fengxiang Zhao, Yaqing Yang, Yibing Chen et al.

This study presents a high-order finite volume scheme capable of large time-step integration for three-temperature radiation diffusion (3TRD) equations, where conservation is naturally achieved through energy update. To handle local large gradients and discontinuities in temperature, a central generalized ENO (GENO) reconstruction is developed for diffusion systems, which achieves essentially non-oscillatory reconstruction for discontinuous solutions. Compared to conventional nonlinear reconstruction methods, its most distinctive feature is the central-type symmetric sub-stencils, which ensure consistency between the numerics and the isotropic nature of thermal diffusion. Additionally, the central GENO method provides smooth states of temperature and temperature gradient at interfaces, facilitating the evaluation of numerical fluxes. Furthermore, interface flux evaluation for cases with discontinuous physical property parameters is modeled. To address the extremely small time-step issue caused by stiff diffusion and source terms, a dual-time-stepping method based on implicit time discretization is developed for the first time in 3TRD systems, with the advantage of decoupling temporal discretization from complex nonlinear spatial discretization. A series of numerical examples validates the high accuracy, physical property preservation, strong robustness, and large time-step integration capability of the present high-order central GENO scheme.