Evaluation of Explicit Numerical Solution Methods of the Muskingum Model

The nonlinear Muskingum model is a frequently used hydrologic routing method. In this model, the rate of change of the storage volume with respect to time is expressed by an ordinary first-order differential equation. Generally, this equation has no analytical solution, and thus, should be solved by standard numerical solution methods. Although many optimization techniques have been employed to estimate the parameters for the model, an accurate solution method for calculating the storage time variation of the Muskingum model is still required. Most previous researchers have used an inaccurate explicit Euler’s method along with a manipulated routing equation for calculating the discharge at the downstream end to achieve a better fit for observed data. This manipulation, however, is not acceptable from a mathematical viewpoint. Until now, the storage time variation of the Muskingum model has only been calculated by an explicit Euler’s method; other explicit numerical solution methods have not been used or clearly discussed. All explicit solution methods may produce similar results for small sized time intervals, but in practice for historical field data, the size of the time interval is fixed and insufficiently small; thus, a suitable solution method with sufficient accuracy should be used. This study proposes a fourth-order Runge-Kutta method as an accurate and suitable explicit solution method for calculating the storage time variation of the Muskingum model.