The present paper is another attempt to make the surface hydrologic community aware of the problems associated with the use of poor numerics. This is always important, and the present hook is climate change (increased precipitation). The work presented in this paper is standard material for a numerics class - how to solve differential equations. I certainly appreciate the topic and second its importance, yet, do like to articulate that a need for more publications on this topic highlights a fundamental flaw in hydrologic education. Proper awareness/knowledge of mathematics, numerics and, also, statistics.

The paper is well written and generally well prepared. Nevertheless, the paper would benefit from a careful read by a native speaker. I will give some examples in my review below. What is more, at a number of different places in the manuscript, I believe the authors do not use the 'right' technical language. I will also provide some examples of this. Altogether, I believe this paper should be published after some moderate, perhaps major revision, if they deem some of the comments important enough and/or worthy of further exploration. I list my comments in random order - as I wrote them down when scrolling through the pdf.    

1. Is it possible to express the numerical error as percentage of the rainfall error? If you make some simple assumptions about the rainfall error. Then you can use this as metric and show how it increases with rainfall intensity/duration. Same question about discharge measurement error. If one makes an assumption about 10% error (heteroscedastic) then one can express the numerical error as percentage of the discharge measurement error. One can show how this error evolves with time. 

2. Then on a related note, can you investigate the relationship between numerical error and flow level (=discharge) ? May be interesting to see - as this, I believe, is not explicitly addressed in earlier studies. This should show that numerical errors are relatively large during rainfall events, and these errors dissipate during a subsequent drying period. This, dissapation is one reason numerics has not got the attention it deserves from the community. I will revisit this point in a later comment.   

3. Did you consider midpoint methods? Why/why not? Similar question for Runge-Kutta methods by the Carl Runge and Wilhelm Kutta? They developed a whole toolbox of explicit/implicit/fixed/variable time step integration methods. What about backward Euler? I know this implies more work, nevertheless, maybe there was a reason not to include these methods in your analysis - then, it would be good to know the reasons. 

4. When it comes to Heun and Euler some papers are cited but the original inventors of these solution methods are not mentioned! I would include a reference to Leonhard Euler (Institutionum calculi integralis) and Karl Heun, among others. 

5. One reason proper numerics receives little attention among surface hydrologists is the nature of hydrologic systems. Negative feedback loops ameliorate differences in initial states and promote convergence to a stable state. Indeed, for such systems one can simply use a spin-up period (as you use in the paper) to prepare the initial states of the model for subsequent simulation. In systems with positive feedback, numerical errors will have a devastating effect on long term behavior as model runs will diverge rapidly and suggest a very different system behavior later on. Thus, one reason that numerical errors have historically received relatively little attention is the nature of hydrologic systems, that is, negative feedback loops induce stable attractor states - hence, why we can solve poor knowledge of the initial states with a spin-up period. Note, in some fields, differential equations are so incredible sensitive to numerics that these small errors can induce chaos (example: two-predator-two-prey systems).    

6. The paper of Schoups et al: doi:10.1029/2009WR008648 draws similar conclusions as herein, that is, the use of a second-order integration methods is preferred. I believe the text should address this earlier paper and those possibly related to it more properly. The paper is cited, but the text does not address that similar findings have been reported elsewhere. On a related note, there are more surface hydrologic model codes that use proper numerics. For example, the hmodel of Schoups et al. (same paper) has been used in various publications. This model uses 2nd order adptive Heun for its numerical solution. 

7. You may want to emphasize in the paper that poor numerics not only affects the simulated discharge, but compromises tasks such as parameter estimation, prediction, simulation of related state variables (groundwater table, soil moisture), etc. Those familiar with proper numerical procedures are aware, but not all those others reading this manuscript.  

8. I do not see anything else that is wrong with this paper (see my written comments on pdf), except that the questions listed above may help find a second hook to 'sell' this work. Climate change is an interesting hook, yet, one may argue that the precipitation intensities as used herein are a bit exaggarated. Maybe a more detailed investigation into how numerical errors behave in a simulation may be interesting, their dependence on simulated flow level and simulated state variables. Perhaps even better, can you pinpoint which process in the model contribute most to the numerical error during precipitation extremes. This must be the most nonlinear process - or that process (its flux) that changes most rapidly in a time step.  

Please see the attached supplement for other comments.

Please find my review attached as a supplement.

Best,

Martina

The manuscript presents an interesting study about numerical errors in hydrological models under extreme precipitation. It assesses several numerical methods with the FUSE framework under different conditions, such as varying initial conditions or precipitation intensity and duration, and different parameter sets and model structures. It shows that numerical errors increase with precipitation intensity, which can become an increasing issue with climate change. Finally, it raises awareness about suboptimal numerical methods often implemented in commonly used hydrological models. 

General comments

The manuscript is well structured, well written, and contains high-quality figures. I have only a few comments and will not address elements already raised by other reviewers. The topic addressed is important, and the focus of the paper on extreme precipitation differentiates it from other articles on numerical errors in hydrological models. Therefore, I recommend publishing the article after corrections.

Specific comments

• The type of hydrological models assessed (i.e., conceptual and lumped) appears very late in the manuscript. I expected to find this information much earlier, and it should even be mentioned in the abstract.
• Numerical methods, l.104: “which in some fields is desirable”. An example could be a nice-to-have small addition.
• Methods: I would expect event durations inferior to 5 days to be relevant for flood modeling in smaller catchments in many places across the World.
• Results, l.306-307: Median (relative) errors of several methods are not always increasing with precipitation intensity as stated (for ex: adaptive explicit Heun, adaptive implicit Heun, adaptive semi implicit Euler). Or have I misunderstood something?
• Results, Section 4.2: This section is clearly the part I like least. First, it requires quite some effort to follow the ranking strategy and the details of the results. Then, the establishment of the groups seems to be a bit arbitrary, with some groups containing two methods and others three, which is also changing between low and high precipitation intensity. Why are there differences in the size of the groups between low and high precipitation? Also, the same colors were used to represent different groups of methods between both panels of Fig. 6, which makes it confusing. Finally, I am not convinced that the conclusion of this analysis is worth the effort it takes to follow all the details of the procedure. I will let the authors judge what they want to do with it, but I would suggest simplifying this section.
• Results, Section 4.3: The impact of the reduced daily intensity with increasing duration is, in my opinion, a likely explanation for the decrease of the errors. I would have thus expected to find this hypothesis earlier in this section.
• Results, Section 4.4: This section would benefit from an introductive or transitional sentence.
• Results, Section 4.7: I agree with Martina Kauzlaric that the link between the models assessed and the numerical methods implemented would benefit the reader.

Technical corrections

• Intro, l.60-61: This sentence might be rephrased for more clarity. This is only a suggestion.
• Intro, l.83-85: It might also be rephrased for more clarity (again, a suggestion).
• Numerical methods, l.137: Citation: the names of the authors should be in the parenthesis.
• Methods, l.197: The model here named 563 should be 536, according to Clark and Kavetski (2010)
• Results, l.474-475: This sentence is not so clear and might be rephrased.