Supplement of Numerical daemons of hydrological models are summoned by extreme precipitation

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation such as that observed during hurricanes Harvey and Katrina
impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation. In this experiment, a large number of hydrographs are generated with the modular modeling framework FUSE (Framework for Understanding Structural Errors), using eight numerical techniques across a variety of forcing data sets. All constructed models are conceptual and lumped. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational cost and numerical error associated with each hydrograph were recorded. Numerical error is assessed via root mean square error and normalized root mean square error. It was found that the root mean square error usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both a small numerical error and a low computational cost. A small literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be suboptimal. We conclude that relatively large numerical errors may be common in current models, highlighting the need for robust numerical techniques, in particular in the face of increasing precipitation extremes.


Here, the numerical methods used by the surveyed modeling codes are detailed. The references which explain the numerical method for each code are listed in Appendix A of the associated manuscript. This literature review was performed in the first half of 2020; codes have possibly been updated since then. In the following, "Category" can be A, B, or C and corresponds to categories of Fig. 10 in the manuscript. Note that Euler methods are first order.
Model: HBV light Category: C Remarks: This model uses the fixed step explicit Euler method, placing it in category C. It uses operator splitting.
Model: dynamic TOPMODEL Category: C Remarks: This model uses a fixed time step, placing it in category C. It uses an implicit method where the default time step is divided into substeps (with a default time step of 1 hour and a default of 4 sub steps per time step). While the small default time step, fixed substepping, and implicit method might be able to constrain numerical errors, this model does not employ adaptive substepping.
Model: GR4J Category: B Remarks: Adaptive implicit Euler has been implemented. This model uses the state-space formulation, rather than operator splitting.
Model: VIC Category: C Remarks: This model uses a fixed time step, placing it in category C. It uses sequential solving, as it uses the fixed step explicit Euler method for some fluxes and the fixed step implicit method for others.
Model: Sacramento SMA Category: C Remarks: This model uses a fixed time step, placing it in category C. Some fluxes use implicit methods and others explicit; therefore this model employs operator splitting.
Model: mHM Category: C Remarks: This model uses the fixed step explicit Euler method for most of the model fluxes, placing it in category C, although its routing routine is adaptive. Therefore it is also operator splitting.
Model: PRMS Category: C Remarks: This model uses a fixed (daily) time step, placing it in category C, and apparently uses an explicit method.
MMF: FUSE Category: A Remarks: FUSE employs simultaneous (rather than operator-splitting) solutions, has first and second order solvers, has implicit, explicit, and semi-implicit methods, and has adaptive substepping. It contains a second order explicit adaptive method.
MMF: SUMMA Category: B Remarks: SUMMA employs simultaneous differential equation solving, has explicit and implicit solvers, and has adaptive substepping. It does not however have a higher order method than 1.

MMF: MARRMoT
Category: C Remarks: Fixed step implicit or explicit Euler methods are available. This MMF employs a state-space (rather than operator splitting) formulation of its equations.
MMF: SUPERFLEX Category: C Remarks: this MMF employs the fixed step implicit Euler method. This MMF employs a state-space (rather than operator splitting) formulation of its equations.
MMF: Raven Category: B Remarks: This MMF is capable of using a second order implicit adaptive numerical method, and hence is place in category B. It does not have a second order explicit adaptive method. It also has the option for operator splitting, where specific numerical methods can implemented per flux, but also has the option for space state formulation. It also has the explicit Euler option.