Robust numerical solution of the reservoir routing equation

• Numerical solutions of the reservoir routing equation are evaluated.
• The Laurenson–Pilgrim method cannot handle nonmonotonic outflow rating curves.
• Runge–Kutta methods may fail when large time steps (greater than 12 min) are used.
• The Cash–Karp method does not overcome the limitations of Runge–Kutta methods.
• Backstepping allows Runge–Kutta methods to be accurate, efficient, and robust.

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson–Pilgrim method, (2) the fourth-order Runge–Kutta method, and (3) the fixed order Cash–Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.