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High order fast sweeping methods for efficiently solving steady statesolutions of hyperbolic PDEs were not available yet on unstructured meshes. Inthis paper, we extend high order fast sweeping methods to unstructuredtriangular meshes by applying fixed-point iterative sweeping techniques to afifth-order finite volume unstructured WENO scheme, for solving steady statesolutions of hyperbolic conservation laws. An advantage of fixed-point fastsweeping methods which distinguishes them from other fast sweeping methods isthat they are explicit and do not require inverse operation of nonlinear localsystems. As in the first order fast sweeping methods on triangular meshes, weintroduce multiple reference points to determine alternating sweepingdirections on unstructured meshes. All the cells on the mesh are orderedaccording to their centroids' distances to those reference points, and theresulted orderings provide sweeping directions for iterations. To make theresidue of the fast sweeping iterations converge to machine zero / round offerrors, we follow the approach in our early work of developing the absolutelyconvergent fixed-point fast sweeping WENO methods on rectangular meshes, andadopt high order WENO scheme with unequal-sized sub-stencils for spatialdiscretization. Extensive numerical experiments are performed to show theaccuracy, computational efficiency, and absolute convergence of the presentedfifth-order fast sweeping scheme on triangular meshes. Furthermore, theproposed method is compared with the forward Euler method and the popular thirdorder total variation diminishing Rung-Kutta method for steady statecomputations. Numerical examples show that the developed fixed-point fastsweeping WENO method is the most efficient scheme among them, and especially itcan save up to 70% CPU time costs than TVD-RK3 to converge to steady statesolutions.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws