Application of different explicit Runge–Kutta methods to solve gradually varied flow equations

Gradually Varied Flow (GVF) is a steady-state flow condition in which water depth varies gradually along the channel. From a mathematical viewpoint, Runge-Kutta (RK) is distinguished for its intense calculations per step, and hence, considered appropriate for GVF depth computations. In this study, performance of various explicit and embedded RK methods to solve GVF equations was investigated. In this regard, sixteen RK methods (first- to fifth-order) were used to determine water surface profiles under GVF conditions. Three hypothetical prismatic channels with different bottom width, side slope, bottom slope, flow rate, known water depth, and resistance coefficient values were considered. Calculated water surface profiles for all 96 scenarios were compared with analytical solutions available in the literature. Results for M1 and S1 profile calculations reveal that most RK methods yield acceptably accurate results, while Euler’s method appeared to be the most efficient one. On the other hand, it was found that Ralston and Kutta s third-order methods perform the worst among all considered methods.