Predicted hydrographs are obtained for the ACC, FV1-CPU, FV1-GPU, DG2-CPU and DG2-GPU solvers (Fig. ). FV1-CPU and FV1-GPU solutions are identical and are named collectively as FV1 (similarly, DG2-CPU and DG2-GPU are named collectively as DG2). While no exact solution is available, DG2, FV1 and ACC predictions of water depth and velocity agree closely with existing industrial model results (Fig. 4.10 and 4.11 in ). ACC and DG2 water depth predictions are almost identical at all gauge points (Fig. a–d). FV1 predictions are nearly identical, except that the wave front is slightly smoother and arrives several minutes earlier than ACC or DG2, as seen at point 5 (Fig. c) and point 6 (Fig. d).

Differences in velocity predictions are more pronounced (Fig. e–h). The biggest differences are seen at point 1 (Fig. e), located only 50 m from the breach, since the flow at this point is dominated by strong inflow discharge with negligible retardation by frictional forces. At point 1, ACC and DG2 velocity predictions agree closely with the majority of industrial models (Fig. 4.11 in ). LISFLOOD-FV1 predicts faster velocities up to 0.5 ms-1, which is close to the prediction of TUFLOW FV1 Table 11. Further away from the breach at point 3 (Fig. f), point 5 (Fig. g) and point 6 (Fig. h), velocity predictions agree more closely, except at the time of wave arrival. At this time, DG2 predicts the sharpest velocity variations while ACC velocity predictions are slightly smoother. FV1 predicts even smoother velocity variations with slightly lower peak velocities.

Spatial grid convergence is studied by modelling at grid resolutions of Δx=5, 1 and 0.5 m. Since the floodplain is flat, no topographic resampling is required. On each grid, the water depth cross section is measured along the centre of the domain (Fig. ). DG2, FV1 and ACC cross-sectional profiles at the standard grid spacing of Δx=5 m agree well with industrial model results (Fig. 4.13 in ). Differences are most apparent in the vicinity of the wave front, near x=400 m. At the standard resolution of Δx=5 m, FV1 predicts a wave front about 50 m ahead of ACC or DG2, and the FV1 solution is much smoother. A TUFLOW modelling study reported similar findings, with TUFLOW FV1 predicting a smoother wave front about 50 m ahead of other TUFLOW solvers Table 12. At a 5 times finer resolution of Δx=1 m, all solvers predict a steeper wave front, although the FV1 wave-front prediction at Δx=1 m is still relatively smooth, being closer to the ACC prediction at Δx=5 m. A 10 times finer resolution of Δx=0.5 m is required for FV1 to predict a steep wave front in agreement with DG2 at Δx=5 m, while ACC only requires a resolution of Δx=2 m to obtain similar agreement.

These differences can be attributed to the order of accuracy of the solvers: DG2 is formally second-order-accurate and exhibits the least sensitivity to grid resolution; FV1 is formally first-order-accurate and exhibits the greatest sensitivity, with numerical diffusion errors leading to a spuriously smooth wave. Despite its simplified formulation, ACC predictions are close to DG2 because ACC is second-order-accurate in space (Sect. ).

To assess the relative runtime cost of the solvers, the test is run for a range of grid resolutions from Δx=10 m (yielding 2×104 elements) to Δx=0.5 m (8×106 elements). Each of the ACC, FV1-CPU and DG2-CPU solvers are run using 16 CPU cores, and FV1-GPU and DG2-GPU are run on a single GPU. To ensure reliable measurements, each solver is run twice on each grid, and the fastest runtime is recorded. Runs that have not completed within 24 h are aborted and excluded from the results. Solver runtimes are shown in Fig. a on a log–log scale.

On the coarsest grid with 2×104 elements, FV1-CPU and FV1-GPU both take 5 s to complete – just 2 s more than ACC. As the grid is refined and the number of elements increases, FV1-CPU remains slightly slower than ACC, while FV1-GPU becomes faster than ACC when Δx<5 m and the number of elements exceeds 105. The runtime cost relative to ACC is shown in Fig. b: FV1-CPU is about 1.5–2.5 times slower than ACC, gradually becoming less efficient as the number of elements increases. In contrast, FV1-GPU becomes about 2 times faster than ACC (relative runtime ≈ 0.5) once the number of elements exceeds 106 (Δx∼1 m), when the high degree of GPU parallelisation is exploited most effectively.

Similar trends are found with DG2-CPU and DG2-GPU: on the coarsest grid DG2-CPU is about twice as fast as DG2-GPU, but DG2-GPU becomes increasingly efficient as the number of elements increases, being twice as fast as DG2-CPU at Δx=2 m with 5×105 elements (Fig. c). At Δx=1 m with 2×106 total elements, DG2-GPU completes in about 3.5 h while the DG2-CPU run is aborted, having failed to complete within 24 h (Fig. a).

As seen earlier in the inset panel of Fig. , similar wave fronts were predicted by DG2 at Δx=5 m, ACC at Δx=2 m and FV1 at Δx=0.5 m. At these resolutions, DG2-CPU, DG2-GPU and ACC achieved a similar solution quality for a similar runtime cost, with all solvers completing in about 4 min (Fig. a). Meanwhile, the DG2 solvers on a 10 times coarser grid were 140 times faster than FV1-CPU (10 h 42 min) and 28 times faster than FV1-GPU (1 h 47 min).

To assess the computational scalability of the multi-core CPU solvers, the test is run using 1–16 CPU cores, while FV1-GPU and DG2-GPU are run on a single GPU device. A grid spacing of Δx=2 m (5×105 elements) is chosen so that the grid has sufficient elements for effective GPU parallelisation (informed by the GPU runtimes in Fig. b–c) but has few enough elements so that all model runs complete within the 24 h cutoff. Measured runtimes for ACC, FV1-CPU and DG2-CPU are shown in Fig.  on a log–log scale, with all solver runtimes decreasing as the number of CPU cores increases. To facilitate a like-for-like comparison with FV1 and DG2, ACC solver runtimes were obtained for the ACC implementation of . Theoretical “perfect scaling” lines are marked by thin dotted lines for each solver: perfect scaling means that doubling the number of CPU cores would halve the runtime. ACC solver scalability falls somewhat below perfect scaling, with a 16-fold increase in CPU cores only yielding a 7-fold decrease in runtime (Fig. ). In contrast, the DG2-CPU and FV1-CPU solvers achieve close-to-perfect scaling up to 4 CPU cores, with synchronisation overheads causing only a small decrease in scalability thereafter. It is expected that additional performance can be gained by using the alternative, CPU-optimised ACC implementation , and these CPU-specific optimisations are also under consideration for future enhancement of the DG2 and FV1 solvers. For intercomparison with the CPU solvers, FV1-GPU and DG2-GPU runtimes are also marked by dashed horizontal lines (since the number of GPU cores is not configurable). Both GPU solvers are substantially faster than their counterparts on a 16-core CPU.

Overall, the FV1, ACC and DG2 solvers converged on similar water depth solutions with successive grid refinement. Owing to its first-order accuracy, FV1 requires a very fine-resolution grid to match the solution quality of DG2 or ACC, though FV1-GPU enables runtimes up to 6 times faster than the 16-core FV1-CPU solver. Thanks to its second-order accuracy, DG2 water depth predictions are spatially converged at coarser resolutions (Fig. ). Hence, DG2 is able to replicate the modelling quality of FV1 at a much coarser resolution, and the multi-core DG2-CPU solver is a competitive choice for grids with fewer than 100 000 elements.

Predicted water level and velocity hydrographs are shown in Fig. . The water level hydrographs show that water ponds in small topographic depressions at point 1 (Fig. a), point 3 (Fig. b) and point 5 (Fig. c). Point 7 is positioned near the steep valley slope and is only inundated between t=1 h and t=8 h (Fig. d). At both resolutions, water levels predicted by all solvers agree closely with existing industrial model results at points 1, 3 and 7 (Fig. 4.16 in ). Small water level differences accumulate as water flows downstream, and at point 5, positioned farthest downstream of the dam break, differences of about 0.5 m are found depending on the choice of resolution and solver (Fig. c). Similar water level differences have been found amongst the suite of TUFLOW solvers and amongst other industrial models . Bigger differences are found in velocity predictions, particularly at locations farther downstream at point 3 (Fig. f), point 4 (Fig. g) and point 7 (Fig. h). At point 3, DG2 predicts small, transient velocity variations at Δx=50 m starting at t=1 h; these variations are not captured by the FV1 or ACC solvers but have been captured by a FV2-MUSCL solver at the finest resolution of Δx=10 m, as reported by . At point 7, ACC overpredicts peak velocities by about 0.5 ms-1 compared to FV1 and DG2 (Fig. h), and compared to other industrial models (Fig. 4.17 in ). Otherwise, ACC, FV1 and DG2 velocity predictions are within the range of existing industrial model predictions.

While hydrograph predictions are often studied for this test case , flood inundation maps taken at time instants provide a more detailed picture of flood wave propagation. Accordingly, two sets of flood inundation maps are obtained: one set at t=15 min during the short period of peak inflow and another set at t=3 h once the flood water has almost filled the valley. Flood maps are obtained at the finest resolution of Δx=10 m and with DG2 at Δx=50 m.

After 15 min, water has travelled about 1.5 km north-east along the valley away from the inflow region near the southern edge of the domain, with ACC, FV1 and DG2 water depth predictions shown in Fig. a–d. Behind the wave front, an abrupt change in water depth is predicted by FV1 (Fig. b) and DG2 (Fig. c, d), but this discontinuity induces spurious, small-scale oscillations in the ACC solver that propagate downstream (Fig. a). This numerical instability is understood by studying the Froude number, as shown in Fig. e–h. The rapidly propagating flow becomes supercritical across the region of shallower water, with a maximum Froude number of around 1.5. The local inertia equations are not physically valid for modelling abrupt changes in water depth or supercritical flows , leading to the observed numerical instability in the ACC solver.

After 3 h, the flood water has filled most of the valley and the wave front has almost reached point 5. As shown in Fig. i–l, ACC, FV1 and DG2 water depth predictions are in close agreement. The flow is now predominantly subcritical (Fig. m–p), although a small region of supercritical flow is found upstream of point 3 with a maximum Froude number of about 1.2 and a corresponding jump in water depth at the same location. Nevertheless, numerical instabilities in the ACC prediction at t=15 min are no longer evident at t=3 h (Fig. m), and ACC predictions remain stable at all gauge points for the duration of the simulation (Fig. ). As seen in the fourth column of Fig. , DG2 flood maps at Δx=50 m are in close agreement with the ACC, FV1 and DG2 flood maps at Δx=10 m.

Simulation runtimes are summarised in Table , with FV1-CPU and DG2-CPU solvers run on a 16-core CPU and FV1-GPU and DG2-GPU solvers run on a single GPU. Similar to runtime measurements presented earlier in Sect. , the GPU solvers become more efficient on grids with a larger number of elements: in this test, DG2-GPU is 1.8 times faster than DG2-CPU at the standard resolution of Δx=50 m, becoming 2.5 times faster at the finest resolution of Δx=10 m; similarly, FV1-GPU is between 1.2 times and 5.1 times faster than FV1-CPU.

DG2-CPU and DG2-GPU at Δx=50 m outperform ACC, FV1-CPU and FV1-GPU at Δx=10 m, while still achieving similarly accurate flood map predictions at a 5 times coarser resolution (Fig. ). DG2-CPU at Δx=50 m is 2 times faster than ACC at Δx=10 m, while DG2-GPU is twice as fast again. DG2-GPU flood maps at an improved resolution of Δx=20 m are obtained at a runtime cost of 38 min, which is still competitive with ACC at Δx=10 m (with a runtime cost of 30 min).

In summary, all solvers predicted similar water depth and velocity hydrographs, though ACC experienced a short period of numerical instability in a localised region where the Froude number exceeded the limit of the local inertia equations. The shock-capturing FV1 and DG2 shallow-water solvers yield robust predictions throughout the entire simulation. with FV1-GPU being consistently faster than ACC on a 16-core CPU. As found earlier in Sect. , DG2 at a 2–5 times coarser resolution is a competitive alternative to ACC and FV1, with the GPU implementation being preferable when running DG2 on a grid with more than 100 000 elements.

Free-surface elevation hydrographs at the 16 river gauging stations are shown in Fig. . Observed free-surface elevation hydrographs are calculated from Environment Agency measurements of water depth and riverbed elevation above mean sea level . While water depths can be measured to an accuracy of ∼0.01 m , discrepancies between in situ, pointwise riverbed elevation measurements and the remotely sensed, two-dimensional DEM can result in systematically biased free-surface elevations, as reported by .

As seen in the observed hydrographs, river levels begin to rise across the catchment around 00:00, 5 December. A flashy response is seen in the headwaters of the river Eden, at Temple Sowerby, Great Musgrave Bridge and Kirkby Stephen, with water levels rising rapidly by 2–3 m and returning almost as rapidly to base flow conditions around 00:00, 7 December. Similar responses are found at the other gauging stations located further downstream, where river levels vary more gradually. The largest river level changes are found in the Carlisle area, particularly at Sheepmount and Sands Centre, which are located farthest downstream.

Timings of the rising and falling limbs are well-predicted by all three solvers for the majority of hydrographs. At coarser grid resolutions, river levels are overpredicted and the difference between base flow and peak flow levels is underpredicted.

This is the case except for anomalous predictions found at Great Musgrave Bridge, where the observed hydrograph shape is generally well-captured but free-surface elevations are consistently underpredicted. This anomaly is due to localised terrain elevation differences between the finest-resolution DEM and Environment Agency riverbed elevation measurements, as documented by .

Maximum flood extents are obtained for ACC and FV1 runs at resolutions of Δx=40, 20, and 10 m; due to its relatively high runtime cost, DG2-GPU is only run at Δx=40 m only with flood maps at 20 and 10 m being inferred by downscaling the 40 m solution. The downscaling procedure adopted here exploits the full, DG2 piecewise-planar solution by constructing the piecewise-planar maximum water depth on the Δx=40 m grid and then sampling at the element centres on the higher-resolution grid.

As shown in Fig. , the post-event survey outlined in pink marks the maximum extent of flooding across Carlisle. The surveyed flood extent is well-predicted by all solvers. Predicted flood extents are largely insensitive to grid resolution, except for the region around Denton Holme gauging station on the river Caldew, which is protected by flood defence walls. added these flood defence walls by hand in the original 5 m DEM, but the coarsened DEMs were upscaled with no further hand-editing. As such, the steep, narrow walls become smeared out at coarse resolutions, with all solvers overpredicting flood extents at Δx=40 m in the Denton Holme region. The representation of these flood defences could be improved by adopting the recently developed LISFLOOD-FP levee module

Not yet available with the FV1 or DG2 solvers.

While some differences between solver predictions were evident in maximum flood depths, these differences become clearer in flood inundation maps taken at a single time instant. Accordingly, flood inundation maps shown in Fig.  are taken at 12:00, 5 December, over Carlisle city centre, during the rising limb of the Sheepmount and Sands Centre hydrographs, where river level rises were largest (Fig. ). At the coarsest resolution of Δx=40 m, DG2 and ACC predictions are almost identical (Fig. a and  g). Both solvers accurately capture the flood extent and water depths predicted by FV1 and ACC at a 4 times finer resolution of Δx=10 m (Fig. f and  i). In contrast, FV1 predicts greater water depths and a slightly wider flood extent, particularly at coarser resolutions of Δx=40 m (Fig. d) and Δx=20 m (Fig. e). But once the grid is refined to a resolution of Δx=10 m, FV1 and ACC solutions are almost converged (Fig. f and i). DG2 predictions at Δx=20 m (Fig. b) and 10 m (Fig. c) are downscaled from the DG2 prediction at Δx=40 m. The downscaled DG2 predictions are not expected to resolve all fine-scale features visible in the FV1 and ACC predictions. Nevertheless, compared to the DG2 Δx=40 m flood map, the downscaled DG2 flood maps better represent the deeper waters in the river Eden (flowing east to west) and in the river Caldew (flowing south to north).

To quantify the spatial convergence of the three solvers, water depth RMSEs are calculated at 12:00, 5 December, over the entire catchment (Table ). Since water depth observations are unavailable, the FV1 prediction at Δx=10 m is taken as the reference solution. At Δx=40 m, DG2 and ACC RMSEs are almost identical, while the FV1 error is about 10 % larger. At Δx=20 m, FV1 errors are again about 10 % larger than ACC, with the ACC solver converging more rapidly towards the FV1 reference solution than FV1 itself, despite ACC's simplified numerical formulation (Sect. ).

In this catchment-scale simulation and earlier in the simulation of a slowly propagating wave over a flat floodplain (Sect. ), FV1 was seen to converge more slowly and predict a flood extent wider than DG2 or ACC. Once again, these differences can be attributed to the order of accuracy of the solvers: FV1 is formally first-order-accurate and exhibits the greatest sensitivity to grid resolution, while DG2 and ACC are both second-order-accurate in space.

Solver runtimes for the entire 5.5 d simulation are shown in Table . On the same grid, FV1-GPU is about 2 times faster than ACC on a 16-core CPU, which is consistent with earlier findings in Sect.  and . FV1-GPU and ACC runtimes scale as expected: halving the grid spacing quadruples the total number of elements and halves the time step due to the CFL constraint. Hence, halving the grid spacing multiplies the runtime by a factor of about 8. At all grid spacings between 40 and 10 m, FV1-GPU and ACC simulations run faster than real-time and complete in less than 5.5 d, indicating that these solvers are suitable for real-time flood forecasting applications.

The DG2-GPU solver runtime is substantially slower than other solvers on the same, coarse grid. Unlike the tests presented earlier in Sect.  and , this test involves widespread overland flow driven by continual rainfall. Overland flow is characterised by thin layers of water only centimetres deep, which continually flow downhill, driven by gravity and balanced by frictional forces. These frictional forces become strongly nonlinear for such thin water layers, and the DG2 friction scheme imposes an additional restriction on the time step size to maintain stability in the discharge slope coefficients. This challenge has recently been addressed in finite-volume hydrodynamic modelling using an improved friction scheme that calculates the physically correct equilibrium state between gravitational and frictional forces . Extending this friction scheme into a discontinuous Galerkin formulation is expected to alleviate the time step restriction and reduce DG2 solver runtimes for overland flow simulations. For simulations without widespread overland flow, including those presented earlier in Sect.  and , the current DG2 formulation imposes no additional time step restriction and DG2 solver runtimes are substantially faster.