ROW Methods Adapted to Network Simulation for Fluid Flow

Simulating free-surface and pressurised flow is important to many fields of application, especially in network approaches. Modelling equations to describe flow behaviour arising in these problems are often expressed by one-dimensional formulations of the hyperbolic shallow water equations. One established approach to realise their numerical computation is the method of lines based on semi-discretisation in space (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001; Steinebach and Weiner, Appl Numer Math 62:1567–1578, 2012; Steinebach et al., Modeling and numerical simulation of pipe flow problems in water supply systems. In: Martin, Klamroth, et al. (eds) Mathematical optimization of water networks. International series of numerical mathematics, vol 162, pp 3–15. Springer, Basel, 2012). It leads to index-one DAE systems as algebraic constraints are required to realise coupling and boundary conditions of single reaches.Linearly implicit ROW schemes proved to be effective to solve these DAE systems (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001). However, under certain conditions an extended partial explicit time-integration of the shallow water equations could be worthwhile to save computational effort. To restrict implicit solution by ROW schemes to stiff components while using explicit solution by RK methods for remaining terms, we adapt ROW method ROS34PRW (Rang, J Comput Appl Math 262:105–114, 2014) to an AMF and IMEX combining approach (Hundsdorfer and Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. Springer, Berlin, Heidelberg, New York, 2003). Applied to first test problems regarding open channel flow, efficiency is analysed with respect to flow behaviour. Results prove to be advantageous especially concerning dynamical flow.