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In this paper, we design a new type of high order finite volume weighted essentially nonoscillatory (WENO) schemes to solve hyperbolic conservation laws on triangular meshes. The main advantages of these schemes are their compactness and robustness and that they could maintain a good convergence property for some steady state problems. Compared with the classical finite volume WENO schemes [C. Hu and C.-W. Shu, J. Comput. Phys., 150 (1999), pp. 97-127], the optimal linear weights are independent of the topological structure of the triangular meshes and can be any positive numbers with the one requirement that their summation is one. This is the first time any high order accuracy with the usage of only five unequal sized stencils in a spatial reconstruction methodology on triangular meshes has been obtained. Extensive numerical results are provided to illustrate the good performance of such new finite volume WENO schemes.

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