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In recent years, the complexity of vessel networks for one-dimensional blood flow models has significantly increased, because of enhanced anatomical detail or automatic peripheral vasculature generation, for example. This fact, along with the application of these models in uncertainty quantification and parameter estimation poses the need for extremely efficient numerical solvers. The aim of this work is to present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on (i) a high-order finite volume type numerical scheme, (ii) a high-order treatment of the numerical solution at internal vertexes of the network, often called junctions, and (iii) an accurate LTS strategy. The accuracy of the proposed methodology is verified by empirical convergence tests. Then, the resulting LTS scheme is applied to arterial networks of increasing complexity and spatial scale heterogeneity, with a number of one-dimensional segments ranging from a few tens up to several thousands and vessel lengths ranging from less than a millimeter up to tens of centimeters, in order to evaluate its computational cost efficiency. The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]