We use two 2D surface flow solvers in this work, namely SERGHEI , which solves the fully dynamic shallow-water equations, and RIM2D , which solves a local inertia approximation. The key advantage of SERGHEI is that it can be deployed on very large-scale HPC systems, leveraging massively parallel scientific hardware. This allows us to offset the comparatively larger computational cost of solving the full shallow-water equations. In contrast, RIM2D allows us to solve the comparatively cheaper local inertia equations, arguably requiring fewer computational resources, albeit in the current version 0.2 limited to a single GPU.

The Ahr river is an 86 km long tributary of the Rhine river, located in the states of Rhineland-Palatinate and North Rhine-Westphalia (Germany) in the Eifel region. Our study domain focuses on the downstream reach of the Ahr river, spanning approximately 30 km between the towns of Altenahr and Sinzig. In the first third of the reach the river valley is still very enclosed but opens upstream to the town of Bad Neuenahr-Ahrweiler into a wider valley floor. The area consists of mostly rural areas, with a handful of small settlements and the comparatively larger urban area of Bad Neuenahr-Ahrweiler (population of approximately 26 500) . The average annual precipitation level of the region is below the German mean at around 675 mm .

The nearly stationary low-pressure system “Bernd” resulted in heavy rainfall events in western and central Europe in mid-July 2021 which triggered severe and sudden flooding especially in Belgium, the Netherlands, and Germany . The Ahr valley was one of the locations in Germany which was severely affected, accounting for overall 70 % of all fatalities in Germany , 189 in the area around the Eifel, making it the second largest water-related disaster in recent history in Germany in terms of casualties. Numerous factors contributed to this extreme impact. Firstly, the Eifel embodies a low-mountain terrain characterised by steep slopes and narrow valleys, extensively settled and cultivated by communities over an extended period. Consequently, the limited space results in a concentration of both population and structures in vulnerable zones. Furthermore, such areas are inherently susceptible to significant issues like mass movement, rapid erosive discharge, and substantial debris accumulation. These conditions notably caused extensive blockages, resulting in the destruction of numerous bridges along the Ahr river in July 2021, exacerbating the flood surge . During the 14 July 2021 event, water levels in the Ahr reached their highest values at the available gauging stations since the beginning of their measurements. Although the exact water levels are unknown, as most gauging stations along the Ahr river were damaged or destroyed during the event, there are estimates of water levels of around 9 m at the Altenahr gauge , where the normal water depths of the Ahr are less then 1 m.

To quantitatively evaluate flood inundation in a domain, a diverse set of metrics are used to identify over- and under-predictions and their proportions. To compute these metrics, the maximum inundation maps of the simulations are evaluated against each other and the observed flood extent. At first, cells are classified with respect to Table . This is done by comparing the simulation results of RIM2D to SERGHEI. In addition, the results of each model are also compared to the observed inundation extent. From each comparison a confusion map is generated. From this map, the total counts of the indices shown in Table  are computed and used to calculate the domain-wide inundation metrics shown in Table . These metrics are adapted from and . It is important to note that when contrasting RIM2D with SERGHEI, the outcomes generated by RIM2D are considered observed results, as indicated in Table 

One of the core questions of this study is whether these solvers are fast enough for their use in early warning systems. Consequently, we first examine the runtime and computational resources required to perform these simulations.

All simulations reported here were computed on NVIDIA A100 GPUs on the JUWELS Booster supercomputer at the Jülich Supercomputing Centre, as well as in the GFZ Linux Cluster.

Figure  shows the absolute (Fig. a) and relative simulation runtimes for RIM2D and SERGHEI across the four resolutions (relative to each other in Fig. b and relative to the event duration in Fig. c). Notably, at coarser resolutions (dx=5 and 10 m), both models result in very short runtimes, clocking in at least 99 times faster than the duration of the 2021 flood event. This level of efficiency renders both models highly suitable for enhancing existing operational flood forecast systems while maintaining exceptional forecast lead times. Consequently, this capability facilitates detailed flood impact forecasting and swift responses.

As resolutions become finer, the differences in runtime between the two models become more apparent. SERGHEI, employing multiple GPUs, results in runtimes up to 6 times faster than RIM2D, which in the current version 0.2 relies on a single GPU. At finer resolutions, i.e. large number of grid cells to be computed, the computational requirements surpass the parallel computing capabilities of a single scientific-grade GPU, necessitating multi-GPU implementations and some HPC capabilities for operational deployment. It is also notable that at the dx=10 m resolution RIM2D does exhibit a slightly faster runtime compared to SERGHEI. This can be attributed to its less computationally intensive formulation, additionally indicating one GPU to be adequate for simulations at that resolution.

In terms of the usability of these models for flood early warning, Fig. c shows that all simulations were faster than the duration of the event. However, this ratio of event duration to runtime varies between 1 and 400 (for RIM2D) and 10 and 300 (for SERGHEI), depending on the resolution. It is also worth noting that the dx=10, 5, 2, and 1 m resolution models each consist of 1.3, 5.5, 34.7, and 139×106 cells, respectively.

It is relevant to highlight that no specific performance optimisation of the models was carried out for this particular case. Such optimisations could include compiler flags and hardware-based optimisations, domain decomposition strategies, and so on. These can potentially reduce runtimes even further, but they are not necessarily generalisable across cases, software stacks, and hardware. Consequently they are not particularly relevant for the objectives of this study. Nevertheless, such optimisation would be required for operational purposes, which would potentially boost performance even further.

Moreover, continued development in the implementation of the solvers will increase computational efficiency (e.g. by implementing a multi-GPU solver for RIM2D or by dynamically balancing the load across GPUs), so this performance is expected to improve.

The flood indicators illustrating the accuracy of both RIM2D and SERGHEI in replicating flooded areas across various simulations are depicted in Fig. . Overall, both models demonstrate commendable performance, achieving high scores across all indicators. Notably, they exhibit relatively similar performance at coarser resolutions, but differences become more pronounced at finer resolutions. For instance, when considering the critical success index (CSI), both SERGHEI and RIM2D yield comparable results with CSI values above 0.94 at dx=5 and 10 m resolutions, whereas at finer resolutions (dx=1 and 2 m), the CSI values drop into the eighties, highlighting more discernible disparities.

These variations at finer resolutions are evident in the error bias (EB) indicator as well. Specifically, in the 1 and 2 m simulations, the scores for the two models diverge significantly, registering low scores of 54.28 and 16.11, respectively. Notably, the hit rate (HR) indicator stands out as an exception, with scores improving with better resolutions. This disparity is primarily attributable to SERGHEI depicting larger flooded areas in the finer-resolution simulations compared to RIM2D, resulting in a lower false negative (FN) value (as indicated in Table ) and consequently leading to a higher HR score.

Figure  shows the difference in maximum depth between both models for all four resolutions. Areas in which only one of the models predicts wet areas are categorised. It is important to recall that this is not the difference in water depths at any particular time but the difference in the maximum depths reached during the entire event (which may be predicted at a different time by each solver; see Sect. ). The comparisons behave differently along the valley and are strongly affected by resolution. The narrower valley upstream of Mayschoss has reaches with very large differences in water depth, with SERGHEI predicting water depths up to 2.4 m higher than RIM2D at dx=10 m and up to 4 m with dx=1 m. Near Rech there is a trend of SERGHEI predicting much lower water depths than RIM2D, with larger discrepancies at coarser resolutions. Conversely, upstream of Dernau SERGHEI again predicts higher water depths than RIM2D, but the differences are much smaller, on the order of ∼0.4 to ∼1.4 m depending on the resolution. The differences in this narrow river valley with high water depths and flow velocities in the simulated flood events are likely caused by the different mathematical foundation of the models. Under these flow conditions the neglected convective acceleration in RIM2D might play a substantial role in the flow dynamics.

Consequently, in Bad Neuenahr-Ahrweiler, where the valley widens and the water depths and flow velocities reduce, the differences are significantly smaller, with a mix of positive and negative differences. In the region around and downstream of Bad Bodendorf SERGHEI tends to predict shallower depths than RIM2D. Additionally, at higher resolution there are more areas which are flooded by SERGHEI than RIM2D than at coarser resolutions. Of particular interest is that going from 5 to 2 m generates additional flooded areas by SERGHEI in Ahrweiler.

Figure  shows the difference in time to maximum water depth (henceforth lag for brevity) between SERGHEI and RIM2D for the different spatial resolutions used, and Fig.  shows the probability density functions of the lag. The lag is computed as follows: for both solvers, the time at which a particular cell reaches the maximum depth during the simulation is registered, and afterwards the difference (lag) between the time obtained by SERGHEI and RIM2D is computed.

There are both positive (RIM2D predicts earlier maximum depths) and negative (SERGHEI predicts earlier maximum depths) lags. Overall, negative lags only occur upstream of Mayschoss in the narrowest part of the river valley. Clearly, the lag mostly increases from upstream to downstream (i.e. delays accumulate downstream). There are some local regions in which this does not hold (e.g. with dx=5 m, between Mayschoss and Rech). The second point is that the lag range reduces with increasing resolution. At 10 m resolution the lags are significant, up to ∼4 h, roughly 8 % of the duration of the event. At 1 m resolution the lag drops to maximums of ∼2 h, roughly 2 % of the event duration.

In the reconstructed water level graph derived from the Bad Bodendorf gauge , the highest water level occurs at 27.75 h after the start of the simulation period (14 July 2021), which is 2.5 h after the peak in the inflow hydrograph at Altenahr. Herein we refer to the time difference between the peak at these two stations as hydrograph lag, and we use this 2.5 h value as a reference. We computed the same hydrograph lag between both points for the simulations and report it in Table . We also compute the difference between the simulated hydrograph lag and the 2.5 h hydrograph lag estimated by the reconstructed hydrographs. Finally, this difference is expressed as an error relative to the reference hydrograph lag.

Table  shows that the hydrograph lag in SERGHEI reduces significantly with increased resolution, whereas the RIM2D hydrograph lag is far less sensitive. For SERGHEI, the lag difference is always positive; i.e. the peak at Bad Bodendorf is simulated later than the reference in SERGHEI. For RIM2D it is the opposite, it is always negative, meaning that RIM2D simulates a faster peak at Bad Bodendorf than the reference. The relative error is rather constant across resolutions for RIM2D, around -40 %, whereas for SERGHEI, as it is very sensitive to resolution, there are very good results at high resolution but rather poor results at 10 m resolution.

These results suggest that the higher-resolution SERGHEI simulations capture better the flood wave advancement, and decreasing resolution increasingly results in underestimates of the flood wave movement. In contrast, RIM2D seems to overestimate the flood propagation speed but is quite insensitive to resolution. It is worth mentioning that optimising each case individually through individual calibration would very likely lead to improved results because simulated flow velocities and arrival times with different resolutions are sensitive to the roughness parameterisation . Our results (together with contextual knowledge from the literature) also suggest that RIM2D may be more sensitive to roughness calibration (which is reasonable since the local inertia simplifications give a somewhat higher weight to the friction model) and that it may be calibrated at a given resolution and results across resolution should improve. In contrast, whereas SERGHEI seems less affected by the lack of calibration, roughness parameters may need to be calibrated for each resolution.

To further explore the difference in the predicted lag beyond a single gauge point, Fig.  shows the probability density function of the lag between the SERGHEI and RIM2D flood envelopes across all four resolutions. Negative values in the graphs indicate that SERGHEI forecasts an earlier peak in maximum depths, while positive values mean that RIM2D predicts an earlier peak. The comparison in the figure is limited to true positive cells (i.e. areas flooded in both models).

Broadly, the trend indicates that RIM2D consistently forecasts earlier maximum depths compared to SERGHEI across all four resolutions (positive lag values), as already hinted by the lags at the Bad Bodendorf gauge point. The lag between RIM2D and SERGHEI is more pronounced at coarser resolutions than at finer ones. As resolutions become finer, these disparities diminish, and the differences tend to converge toward zero.

The comparisons shown in Fig. , in Fig. , and with the reconstructed flood hydrograph at the Bad Bodendorf gauge imply that RIM2D simulates faster flood wave propagation speed, which is insensitive to the model resolution. The insensitivity to model resolution can be seen positively. The overestimation of the flood propagation speed, however, needs to be considered when interpreting the results particularly in operational flood response if this roughness parameterisation is used. SERGHEI simulates a flood propagation in line with the reconstructed hydrograph at 1 m resolution and tends to underestimate it with increasingly coarser resolutions. This again is also worth considering when using the model at a particular resolution with the presented roughness parameterisation for a particular purpose.

While studies like and suggest that the local inertia approximation results in slower flood propagation speeds compared to the full dynamic equations, it is important to note that Fig.  solely depicts the variance in time to reach maximum water depth, which integrates additional processes and not only wave propagation phenomena. Therefore, we argue that this lag disparity should not be construed as a metric for wave propagation. In the evaluation of the flood propagation simulation it is also worth noticing that the reconstruction of the flood hydrograph at Bad Bodendorf is also a hydrodynamic modelling result and thus also prone to errors in terms of water depths and timing and is thus not an absolute quantitative reference for the evaluation of the model results.

Figure  shows a scatter plot comparison of field observations versus simulated maximum water depths at recorded post-flood observations of maximum water marks on buildings. Figure  shows these same points explicitly in space and colour-codes the relative difference between observations and simulations.

Figure  shows a slight general trend to under-predict rather than over-predict water depths across the domain. Most importantly, it allows us to see how the comparison shifts with resolution. R2 values clearly overall deteriorate for the coarser resolutions. SERGHEI trends towards underestimating more with coarser resolutions, but the opposite is the case for RIM2D.

RIM2D shows closer agreement with observed water marks with coarser-resolution models. As resolution increases, the discrepancy between simulated and observed depths becomes more pronounced. This is particularly clear from Fig. , where it is possible to see how for the 1 and 2 m resolutions RIM2D tends to under-predict maximum depths. At 10 m resolution, there are a few RIM2D points which greatly overestimate observations.

This discrepancy primarily stems from RIM2D's tendency to generate smaller flooded areas in higher-resolution set-ups compared to coarser ones, resulting in under-predicted depths or missing inundation in areas further from the main river channel (see Sect.  for details), which corresponds to the broad behaviour of the points in space (Fig. ).

In contrast, for SERGHEI, a distinct trend among the four resolutions is not apparent, and all model configurations tend to produce deviations within a similar range, arguably with better estimations at higher resolution (Fig. a). Some additional insights can be drawn by also accounting for the spatial distribution of the comparison points, as shown in Fig. . There is a somewhat improving trend towards higher resolution, in which the points located farther from the main river channel which predominantly exhibit under-predicted water depths somewhat improve. In certain cases this is because the predicted inundated area falls short of the location of the points.

Close inspection of the location of the recorded water marks shows that many of the predicted points with the lowest scores, especially at coarser resolution, are the result of poor representation of the buildings in the computational grid. This is illustrated in Fig. , where it can be seen that for a coarse resolution (e.g. 10 m) many of the observation points fall in cells which are identified as buildings, although the point itself is not in the building. As resolution increases more of these points fall into valid areas of the computational domain. This is likely to happen since these observation points are often water marks on walls or urban furniture close to buildings. It is important to highlight that these building representation challenges are present in both the RIM2D and SERGHEI simulation scenarios. To offset this issue, we also allow a search for valid (non-building) cells adjacent to the cell containing the observed point. This allows some leeway to capture more points in the analysis. Moreover, aside from the issues relating to observed points, Fig.  highlights the effect that resolution can have on properly capturing the complex urban environments, even in a fully inundated area as shown in this image. It highlights that although overall metrics may suggest that the 10 m resolution is sufficient to broadly capture the flood, the inundation dynamics in complex urban built-up areas are prone to errors with raster resolutions too coarse to match the urban complexity. In the presented case study in small towns and villages this appears to be the case at 10 m resolution, but this might be different in other urban fabrics.

Figure  shows a comparison of the evolution of flooded areas with increasing water depths for both solvers and all resolutions. Complementarily, Fig.  shows the fraction of flooded area larger than a certain depth threshold relative to the full extent of the flood. In physical terms, Fig. f corresponds to the very deeply flooded areas and of course includes the main channel. This is of course a rather small fraction of the flooded area (less than 10 % for most of the simulations, as shown in Fig. ). In contrast, Fig. a reflects most of the flooded area (only excluding areas flooded with less than 5 cm of water). Arguably, water depths below 10 cm only reflect an inconvenience in terms of flood impact. However, depths of around 50 cm already include flooded underground and ground floors in buildings, have transport potential to move unsecured objects, and represent a danger to human life. A very large fraction (between ∼80 % and ∼90% at the peak) of the flooded areas is indeed flooded with more than 50 cm of water, and up to around 40 % to 50 % of the flooded area exceeds 2 m of depth. This strongly underlines the considerable impact of this flood event.

In comparative terms, the flooded-area evolution for all water depths shows similar behaviours, especially in terms of interpreting the results for flood impact and warning. Nonetheless, some deeper reading of the differences proves insightful.

The SERGHEI simulations show a clear trend of decreasing peak flooded areas and delayed peaks with coarser resolution. This is expected and consistent with well-known hydrograph attenuation and delay due to numerical viscosity (diffusion) .

Interestingly, for RIM2D the effect of resolution is the opposite as in SERGHEI. The local inertia solution results in higher peak areas for coarser resolutions. Additionally, no significant delay of the peaks is observed in the RIM2D flooded-area curves. The insensitivity in the timing to resolution is consistent with the behaviour of diffusive wave (zero-inertia) formulations as discussed in Sect. , and these results suggest that the local inertia approach keeps this property. It is possible that roughness calibrations could alleviate this issue.

Comparing across solvers for the same resolution shows that (i) for the coarser grids (5 and 10 m) SERGHEI results in smaller flood extents than RIM2D across all depth thresholds, (ii) for the finer grids (1 and 2 m) SERGHEI results in larger flood extents than RIM2D across all depth thresholds, and (iii) the peak of the flooded-area curves is somewhat earlier for RIM2D than for SERGHEI for all resolutions. Observations (i) and (ii) are explained by the previous discussion on the effects of resolution on the different solvers.

Observation (iii) is consistent with the discussion in Sect. . Although the lack of convective terms in the local inertia equation typically leads to slower wave propagation in comparison to the full shallow-water equations , this is not reflected in Fig. . It is likely that the complex dynamics of wave propagation and flood buffering in the channel and floodplains may play a more significant role than the attenuated wave propagation speeds. Moreover, as noted by , the relevance of this wave slowdown is greater for higher Froude numbers. In this event, the simulations show that most of the flow field experiences sub-critical conditions (in fact, mostly with Froude<0.6). This suggests that the wave slowdown in the local inertia solver may not be very significant except for very local areas with higher Froude numbers.

Another important aspect in the evaluation and discussion of the simulation results above is that all simulations used the same set of roughness parameters. Many studies e.g. emphasised that roughness used in surface flow solvers is not absolute but has to be regarded as effective roughness. This means that roughness is the main calibration parameter for hydraulic models, which can compensate for effects of model formulations (solvers) and model set-ups (resolution) on simulation results . With dedicated calibrations of both RIM2D and SERGHEI models for different resolutions, it can be expected to reduce the differences in the model results. However, this is out of the scope of this study, which aims at exploring the differences in simulation results caused by solvers and spatial resolution in 2D hydrodynamic models using standard roughness values, just as a modeller might do when exploring potential floods in a new setting and context (precisely what our simulation exercise was intended to represent). A comprehensive study of the sensitivity of roughness coefficients on the RIM2D local inertia solver is expected as future work.