High order fast sweeping methods for efficiently solving steady state solutions of hyperbolic PDEs were not available yet on unstructured meshes. In this paper, we extend high order fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils for spatial discretization. Extensive numerical experiments are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler method and the popular third order total variation diminishing Rung-Kutta method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 to converge to steady state solutions.

翻译：高效求解双曲偏微分方程稳态解的高阶快速扫描方法在非结构化网格上尚不可用。本文通过将定点迭代扫描技术应用于五阶有限体积非结构化WENO格式，将高阶快速扫描方法扩展到非结构化三角网格，用于求解双曲守恒律的稳态解。定点快速扫描方法区别于其他快速扫描方法的一个优势在于其显式特性，无需对非线性局部系统进行逆运算。与三角网格上的一阶快速扫描方法类似，我们引入多个参考点来确定非结构化网格上的交替扫描方向。网格中所有单元按其质心到这些参考点的距离排序，所得排序结果提供迭代的扫描方向。为使快速扫描迭代的残差收敛到机器零位/舍入误差，我们沿用早期在矩形网格上开发绝对收敛定点快速扫描WENO方法的思路，采用具有不等长子模板的高阶WENO格式进行空间离散。大量数值实验验证了所提出的五阶三角网格快速扫描格式的精度、计算效率和绝对收敛性。此外，将所提方法与正向欧拉方法和流行的三阶总变差递减龙格-库塔方法进行稳态计算对比。数值算例表明，所开发的定点快速扫描WENO方法是其中最高效的格式，尤其相较于TVD-RK3可节省高达70%的CPU时间成本以达到稳态解。