Technical note: Quadratic Solution of the Approximate Reservoir Equation (QuaSoARe)

Abstract. This paper presents a method to solve the reservoir equation, a special type of scalar ordinary differential equation controlling the dynamic of conceptual reservoirs found in most hydrological models. The method, called the “Quadratic Solution of the Approximate Reservoir Equation” (QuaSoARe), applies to any reservoir equation regardless of its non-linearity or the number of fluxes entering and leaving the reservoir. The method is based on a piecewise quadratic interpolation of the flux functions, which leads to an analytical and mass-conservative solution. It is applied to two routing models and two rainfall–runoff stores that are representative of hydrological model components, and it is evaluated based on six catchments in eastern Australia that experienced one of the most extreme floods in recent Australian history. A comparison of the method against two standard numerical schemes, the fifth-order Radau implicit Runge–Kutta scheme and the explicit Runge–Kutta scheme of order 5(4), suggests that it can reach similar accuracy while reducing runtime by a factor of 10 to 50 depending on the model considered. At the same time, the method is simple enough to be presented as a short pseudo-code, which is included in this paper. Beyond solving a given reservoir equation, the method constitutes a promising avenue to define flexible models where flux functions are defined as piecewise quadratic functions, which can be solved exactly with QuaSoARe.