In this paper, a third-order weighted essentially non-oscillatory (WENO)scheme is developed for hyperbolic conservation laws on unstructuredquadrilateral and triangular meshes. As a starting point, a general stencil isselected for the cell with any local topology, and a unified linear scheme canbe 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 smoothtests. In the current scheme, an optimization approach is given to deal withthe very large linear weights, and the splitting technique is considered todeal with the negative weights obtained by the optimization approach. Thenon-linear weight with a new smooth indicator is proposed as well, in which thelocal mesh quality and discontinuities of solutions are taken into accountsimultaneously. Numerical tests are presented to validate the current scheme.The expected convergence rate of accuracy is obtained, and the absolute valueof error is not affected by mesh quality. The numerical tests with strongdiscontinuities validate the robustness of current WENO scheme.
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