Scalable Algorithmic Methods for Simulating Heavy-Rain Events

Throughout human history, natural disasters have been a significant threat [17]. Both frequency and extent of heavy-rain events have increased in recent time, as observed by [10] in 2020 for Germany, Thus, the development of efficient and stable tools for hydrodynamic modeling has been an active field for decades, both in research and in product development [12]. The use of high-resolution elevation models is important for capturing terrain details that can have a major impact on the runoff behavior of surface water [11]; however, the high-resolution simulation is computationally highly intensive, making it difficult to employ it for larger areas, especially in early warning systems [13, 8]. Accordingly, many existing tools offer different compromises between modeling details, preprocessing time, and computation time. Scalable approaches for these important tasks are based on combining a number of algorithmic tools, including methods from computational geometry. While previous success stories (most notably by Arge et al. [4, 14, 3, 6, 5, 1, 2, 7]) have made significant progress on stationary flood levels in largely flat terrain, the problem of flooding with time-dependent water flow in sloped terrain remains computationally demanding even with modern tools and commercial software.

Beyond all these efforts for scalable simulation methods, a second, potentially even bigger challenge looms: How can we deal with the unreliable nature of one of the most crucial inputs for any such model – the precipitation data, in time and space? Acquiring this time-dependent input with an accuracy similar to that of a finely resolved, static terrain is highly challenging, and even more difficult when it comes to forecasting such precipitation. As a consequence, computational methods from meteorology, computational physics or modern AI will likely continue to struggle to obtain reliable, fine-grained input, due to the instability of critical weather phenomena. This also implies that real-time intervention, as well as preemptive measures, will continue to suffer from a lack of reliable forecasting of the ensuing runoff.

While our methodological work is applicable to any region with sufficiently detailed terrain data, the main focus area of our work is a $370{\mathrm{km}}^2$ polygonal domain centered on Goslar, Germany, and lies entirely within the federal state of Lower Saxony (Niedersachsen). Elevations increase from ca. $100\mathrm{m}$ above sea level (a.s.l.) in the Oker valley north of Goslar to almost $800\mathrm{m}$ a.s.l. in the southern sector according to Copernicus EU-DEM v1.1 [9].

Testing, improving and benchmarking of our simulations was performed with the commercial solver TUFLOW HPC version 2025.1.0, which is a 1D/2D hydraulic modeling software that solves the depth-averaged Navier-Stokes equations, also known as the shallow water equations, to simulate the flow of water over a terrain. TUFLOW uses an explicit finite volume solution with a square grid, whose cells might vary in size. Further details on the numerical scheme can be found in the TUFLOW manual [16] and in our full paper [15].

In general, there are two main challenges when simulating large areas with high resolution. Firstly, the memory footprint of the simulation becomes prohibitively large, making it infeasible to run the simulation on current hardware. Secondly, the runtime of the simulation also eventually grows beyond a reasonable time frame. This is often countered by using coarser resolution models, which can lead to a loss of accuracy in crucial areas.

Our work [15] is based on two algorithmic improvements. Firstly, we have developed a subdivision algorithm that partitions the terrain into smaller regions that can be simulated independently, thus reducing the memory footprint of the simulation; see Figure 1. Secondly, we use a quadtree refinement approach that (given areas and desired resolution as inputs) automatically refines the mesh in critical regions. A key contribution of our work is the analysis of what regions require refinement and how much refinement is necessary to maintain accuracy while minimizing computational cost; see Figure 2. Both approaches can be combined to enable large-scale simulations on modern hardware.

There are two main outcomes of our work: (I) progress for simulating time-dependent, high-resolution runoff in large-scale heavy-rain scenarios, based on the two algorithmic approaches described in Section 3. In combination with state-of-the-art commercial software and powerful computing devices, these approaches can achieve simulation times that are significantly shorter than the duration of the simulated events for modestly sized regions (on the order of tens of square kilometers) at meter-scale resolution. The exact times depend on many factors, such as the amount of precipitation, condition numbers, terrain characteristics and the presence of 1D model elements such as culverts, which are also modeled. For very large domains (such as our area of interest of $370{\mathrm{km}}^2$ in the mountainous Harz region) at $1\mathrm{m}$ resolution, the limitations of available GPU memory prohibits running a simulation on the entire region at once. Our subdivision technique makes such a simulation feasible, requiring roughly three weeks to simulate a five-hour heavy-rain event on a single machine and eleven days when parallelization is used to the full extent possible with the given dependency graph. Combining this with our grid refinement technique reduces the required time to three days on a single machine and to around $36$ hours when parallelized, see Figure 3.

This progress enables us to demonstrate how the second challenge can be addressed: (II) We use our new computational capabilities to conduct a large-scale, multifaceted sensitivity analysis, providing insights into different scenarios for possible time-dependent runoff and flooding when varying the input precipitation for our study area. This makes it possible to identify both regions of stability (for which the outcomes are largely unchanged under input variations) as well as critically unstable locations (for which the outcome may change significantly under input variations). In particular the latter is of crucial value for effective disaster mitigation: Knowing where relatively small changes can make a significant difference in the outcome is highly valuable when deciding where to invest limited resources with the greatest impact. We demonstrate the viability of this approach by identifying such a critical location for our study area, see Figure 4.

After introducing the global challenges and previous work from Computational Geometry (culminating in SCALGO), the video describes the challenges and approaches to dealing with time-dependet, dynamic scenarios, such as in our study region in the Harz mountains. After visualizing the two main outcomes from above, we highlight how these improvements make it possible to identify critical locations, where small changes can have a large effect.