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Flow direction algorithms have important application in attracting geomorphic features and topographic attributes, which serve as inputs for some hydrological and topographical models. Evaluating flow direction algorithms is of great significance and often conducted on synthetic surfaces instead of real digital elevation models for free of approximation errors. However, most widely-used synthetic surfaces are too simplified to represent complex topographical relief of real-world terrains. For this, this work applies a complex synthetic surface of modified Himmelblau's function (HF) to simulate sophisticated terrains encountered in real landscapes. HF surface is spatially smooth and continuous with four hilltops and one valley, where plan curvatures are clustered mainly from −0.1 to 0.1. In addition, a slope line-based discretization numerical (SLDN) approach is designed for obtaining numerical solution to theoretical total contributing area (TCA) on synthetic surfaces of non-integrable slope lines (e.g. HF surface). TCAs estimated by several flow direction algorithms are compared with SLDN-derived TCA quantitatively. Results indicate that the largest and smallest mean size errors are obtained by Random eight-node (Rho8) (i.e. 82.6 %) and Freeman multiple flow direction (FMFD) (i.e. 17.9 %), while the largest and smallest mean extent errors by Eight drainage directions (D8) (i.e. 128.2 %) and Eight drainage directions, least transversal deviation (D8-LTD) (i.e.55.4 %). Most mean errors are larger than 20.0 %, which may not be satisfactory in practice. This work can provide a reference for flow direction algorithms application in digital terrain analysis, and therefore improve accuracy of hydrological, geological and geomorphological models. [Display omitted] • Characteristics of the complex HF surface are comprehensively investigated. • An SLDN approach is used to obtain theoretical TCAs on complex synthetic surfaces. • D8-LTD and FMFD are respectively the best SFD and MFD algorithms on HF surface. [ABSTRACT FROM AUTHOR]