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 “Quadratic Solution of the Approximate Reservoir Equation” (QuaSoARe) is applicable 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 lead to an analytical and mass conservative solution. It is applied to two routing models and two rainfall-runoff stores that are representatives of hydrological model components and evaluated on six catchments located 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 Radau fifth order implicit and Runge-Kutta of order 5(4) explicit schemes 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 model code is simple enough to be presented as a short pseudo-code included in our 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.