The hydrodynamic model characterizes surface flow through the hyperbolic 2D SWE system, which is based on mass and momentum conservation : 1∂h∂t+∇(hv)=r-i+e+Sw,2∂hu∂t+∂∂xu2+12gh2+∂(huv)∂y=-ghSfx-gh∂zb∂x,3∂hv∂t+∂∂yv2+12gh2+∂(huv)∂x=-ghSfy-gh∂zb∂y, where h is the water depth [L], v=(u,v) is the velocity flow vector [L T⁻¹], r is the precipitation rate [L T⁻¹], i is the infiltration rate [L T⁻¹], e is the evaporation rate [L T⁻¹] and Sw denotes the flow rate [L T⁻¹] provided by sources or sinks. The gravitational acceleration is denoted by g [L T⁻²], zb is the bed elevation [L], and Sfx and Sfy are the friction slope terms, defined as 4Sfx=n2uu2+v2h4/3,Sfy=n2vu2+v2h4/3, where n is Manning's roughness coefficient [T L^−1∕3] .

In a Lagrangian model of material transport, material elements are entrained and transported based on hydrodynamic forces, computed with appropriate coefficients or following a kinematic approach. In this study, the transported particles act as passive tracers: they have negligible volume and mass, do not interact with each other and do not affect the flow field.

In the absence of turbulence, particle transport is driven by advection, and the particle trajectories are governed by the following system of equations that defines the position of each particle xp=(xp,yp,zp): 5dxpdt=u(xp),dypdt=v(xp),zp=h(xp)+zb(xp).

As observed, the vertical position is unaffected by the flow velocity due to the depth-averaged SWE approximation, which neglects the vertical velocity component. Consequently, the particle is assumed to reside at the free surface, computed as the sum of the water depth h and the bed elevation zb. This assignment is not derived from the governing equations but serves primarily to maintain numerical robustness and for visualization purposes. Since the hydrodynamic model is vertically averaged, and thus does not resolve vertical flow structure, particles transported by it do not possess a true vertical coordinate in the physical sense. However, assigning them a position at the free surface ensures consistency with the surface flow and avoids issues arising from irregular bathymetry. Notably, this choice prevents numerical artifacts, such as particles unrealistically crossing obstacles or walls due to inconsistent vertical velocities. Moreover, because the advection velocity is evaluated at the horizontal location of each particle, aligning all particles to the free surface provides a coherent reference for computing motion in the horizontal plane.

Turbulence effects in a flow can be significant and, by modifying the velocity field, also influence particle transport . Therefore, if turbulence is not incorporated into the flow velocity, as it is in the hydrodynamic model of the SERGHEI framework , it can be included in the particle velocity, as follows: 6dxpdt=u(xp)+udisp(xp),dypdt=v(xp)+vdisp(xp),zp=h(xp)+zb(xp), where vdisp=(udisp,vdisp,0) is the velocity induced by the dispersion, where it is considered a null dispersion velocity in the vertical coordinate because of the lack of the advection velocity in the vertical. In this study, a random-walk model is employed to simulate dispersion because of its simplicity and its great balance between computational efficiency and accuracy . This model proposes the vdisp as a function of the diffusivity coefficients Khx and Khy [L² T⁻¹] in the x- and y-coordinates, respectively. In contrast to imposing constant values based on empirical results that depend on the size and shape of the object, the diffusivity coefficients Khx and Khy are here derived from a velocity-dependent expression. Therefore, the standard anisotropic diffusion model is utilized to determine the diffusivity variables, with the following expressions: 7Khx=ϵL|u*|h(xp,yp),Khy=ϵT|u*|h(xp,yp), where ϵL, and ϵT are the longitudinal and transversal dispersion coefficients [L² T⁻¹] , respectively; and |u*| is the friction velocity: 8|u*|=ngvx2+vy2(h(xp,yp))1/3.

To obtain the temporal and spatial evolution of the particles, it is necessary to discretize the system of Eq. () when turbulence is not considered, or Eq. () when it is included. In passive particle transport, particles are considered to have negligible mass and volume, ensuring no interaction with each other or the flow, thereby maintaining unaltered flow properties such as density or viscosity. The choice of discretization method depends on the required accuracy. In this study, two methods are utilized to discretize the equations governing particle trajectories: the forward Euler method and the Runge–Kutta method.

The transport of the particle is governed by the advection. The trajectory can be determined directly from the initial cell to the final cell (see red dashed trajectory in Fig. ), or the particle can be moved cell by cell (see green dotted trajectory in Fig. ). The latter approach is compulsory for addressing potential dry–wet interface issues and for preventing particles from traversing obstacles, as shown in Fig. b.

Thus, the algorithm that loops over the particles must be carefully designed to ensure that each particle traverses each computational cell discretely, dynamically adopting the local velocity and updating its position based on the discrete flow velocities within each cell.

For that, let us consider a single particle p defined by its position xp in the same position of the point p=(px,py), which is inside cell i, with some velocity vi. Let d be the distance between the particle and the point at which the particle would leave the cell when following the velocity within the cell, as shown in Fig. a. In this figure, A=(Ax,Ay) and q=(qx,qy) are the lower-left and lower-right corners (respectively) of the cell, b is the distance between q and the intersection edge point and e^ is the unit vector of the Cartesian y-axis. The distance d is the minimum distance in the direction of the velocity vector v from the point p until it crosses an edge that separates one computational cell from the next. This distance is measured along the line defined by the unit vector v^.

To guarantee that the particle is not advected beyond cell i with the velocity vi, it follows that a geometric restriction for the time step Δtp of particle p is given by 14Δtp≤dvi. This is fundamentally a Courant–Friedrichs–Lewy condition for the particle advection. An additional restriction applies for Δtp related to the hydrodynamic time-step size Δt: 15Δtp≤Δt, ensuring the particle model cannot advance further in time than the hydrodynamic field. When Δtp<Δt, particles must evolve through the full hydrodynamic time step via an iterative sub-cycling process. This involves calculating successive particle time steps Δtp,m until the total advancement matches the hydrodynamic time step (Fig. b): 16Δt=∑m=1MΔtp,m, where M is the number of particle sub-steps within the hydrodynamic time step). This approach can produce situations like those in Fig. , where a particle traverses one, two or (at most) three cells during a single hydrodynamic time step Δt. Since particles can travel through at most three cells, the number of subdivisions M satisfies 1≤M≤3. The value of M is determined individually for each particle based on the flow properties in the cells it traverses: M=1 if the particle remains within the initial cell (Fig. a), M=2 if it crosses into a neighboring cell (Fig. b), and M=3 if it travels through three cells (Fig. c), though this case is uncommon as it requires a specific combination of particle location and velocity field characteristics.

The complete formulation for the particle time-step restrictions is 17Δtp,m≤mindvi,Δt-∑m=1M-1Δtp,m, where i refers to the current cell containing the particle, which may change during the hydrodynamic time step Δt.

The time-step restriction problem is therefore reduced to determining the distance d to the edge through which the particle will exit the cell.

The value of d is calculated as follows:

Given a line with direction v^=vv-1 passing through particle point p, find its intersection with the lines defining the edges, and compute the distance d from p to the four intersection points: north, south, east and west intersections (see Fig. ):18p+dv^=q+be^.Considering only the x-coordinate:19px+dvx^=qx,and therefore, if vx^ is not null,20d=qx-pxvx^=Ax+Δx-pxvx^,where Δx is the cell length in the x-coordinate, which, being a square cell, is equivalent to the length in the y-coordinate. For cases where vx^ is null or negative, the distance d calculated using Eq. () does not yield the correct intersection. For this reason, three additional distances d are calculated by employing a procedure analogous to that used for deriving Eq. ():21d2=Ax-pxvx^if and only ifvx^≠0,22d3=Ay+Δx-pyvy^if and only ifvy^≠0,23d4=Ay-pyvy^if and only ifvy^≠0.

Having computed d, it is possible to evaluate the particle sub-cycling time step (Eq. ), and with that time step advect the particle with Eq. (). This algorithm is used in both the Euler and Runge–Kutta methods.

Thus, the temporal evolution of the x- and y-coordinates of particles are obtained by the proposed algorithm. With respect to the z-coordinate, assuming a passive particle transport with negligible mass and volume for the particles, the value of the water surface level of the cell where the particle is located is imposed, such that it is floating in it. This approach and the use of the algorithm automatically resolve issues such as: (i) the dry–wet problem, provided the numerical method accurately addresses these conditions for the flow; and (ii) wet–wet problems involving topographical changes, where a particle floating at an arbitrary vertical position within a cell may encounter an adjacent cell with a higher elevation, preventing its movement into that cell despite both cells containing water. This ensures that particle movement is physically feasible and adheres to the elevation constraints of the cells.

Introducing a turbulent dispersion term can cause issues in wet–dry cases, as a particle could find itself in a wet cell due to advection and then reach a dry cell when the dispersion term is applied. Therefore, a restriction is imposed: if the destination cell (determined by adding the dispersion term) is dry, the dispersion term is not applied. This restriction is implemented for several reasons:

It is physically inconsistent for a particle with negligible mass to enter a dry cell. The underlying principle is that particles with significant mass could use inertia to traverse dry regions. Therefore, this restriction specifically applies to massless or negligible-mass particles.

Depending on the desired accuracy of the results, different time intervals are proposed to update the evolution of the particles. When the particle advection is computed on every time step of the hydrodynamic solver (from n to n+1), this configuration is referred to as the online mode. In contrast, when particle positions are updated every nIter hydrodynamic time steps (from n to n+nIter), the configuration corresponds to the offline mode. The online mode is expected to yield higher accuracy than the offline mode, but potentially with a higher computational cost due to more frequent advection updates. The explicit Euler method is unsuitable for offline mode due to its limited accuracy, and is therefore applied exclusively in the online configuration. Conversely, the Runge–Kutta method demonstrates sufficient accuracy to be employed in both online and offline modes.

The LPT model has been implemented in the SERGHEI framework, as one of the several modules that have been implemented in this framework (see Fig. ).The SERGHEI framework is a modular and open-source simulation system designed to target performance portability across HPC platforms and standard workstations . It focuses on future-proofing and sustainability while consolidating over a decade of mature numerical techniques for shallow water modeling. SERGHEI facilitates a modular, HPC-ready community model, enabling broader collaboration. The framework moves beyond classical fluvial flooding and hydrograph generation to address applications related to landscape function, transport and integration into Earth system models.

The LPT model has been developed along the same lines as the other modules in SERGHEI. In this way, this model has been implemented in C++ and using the Kokkos performance-portability layer, which allows the simulation of the model using any GPU architecture. As can be seen in Fig. , the LPT model is updated after updating the hydrodynamic model, as it needs the updated information of the conserved variables. The LPT model is currently being implemented to support distributed computations both on multiple CPU nodes (note that shared memory CPU parallelization is achieved via OpenMP) and on multi-GPU systems, following the approach used in other SERGHEI modules.

Initially, a steady circular vortex test case is simulated to validate the model's accuracy. The vortex has a diameter of 10 m and is centered in the middle of the domain. Its velocity field increases radially from zero at the center to a maximum of 100 m s⁻¹ at the outer edge. A steady-state configuration is assumed, meaning that particles released into the flow should maintain a constant radius throughout their trajectories . To ensure reproducibility, the particle is released at a distance of 10 m from the vortex center. The circular vortex is simulated with varying cell sizes, ensuring the particle completed four full revolutions for consistency in error measurement. The vortex radius is consistently set at R=100 m, with a water depth of h=0.1 m and no roughness (n=0). For this test, the hydrodynamic solver is turned off, so that the Eulerian flow fields (water depth and velocity) remained constant and are not evolved over time, also resulting in a constant time step Δt throughout the simulation. Turbulent diffusion terms are excluded from the particle trajectory equations because this is an analytical case, and the objective is to guarantee correctness and robustness of the advection. The Runge–Kutta offline method is updated every five hydrodynamic time steps, a choice made to balance computational efficiency and accuracy. Preliminary tests with shorter update intervals (two to three time steps) yielded only marginal cost savings and limited improvement in accuracy, so the five-step update was selected to better illustrate this trade-off.

The errors for each grid resolution using different numerical configurations are shown in Fig. . The L1-norm for the error at each time step is defined by 24L1(tn)=|xpn-x^|, where xpn is the particle position at the time tn and x^ is the analytical position. The L1-norm integrated in time is the mean absolute error (MAE), which is obtained using 25MAE=1N∑n=0NL1(tn), where N is the number of time measurements.

In addition, the L2-norm at each time step is defined as 26L2(tn)=(xpn-x^)2, while the root-mean-squared error (RMSE) is computed as 27RMSE=1N∑n=0NL2(tn).

Figure shows that, under steady flow, the error of the Runge–Kutta method is similar to that of the Euler method. The algorithm implemented provides a reduced error for the Euler method because of the calculus of the trajectory in each cell (see Fig. ) and in each time step. Moreover, the error of all methods depends strongly on the discretization of the domain, with lower errors for smaller cell sizes. Figure  also shows the error reduction in the same order as the Euler method as the cell size is reduced (order one). However, for the Runge–Kutta method (order four), the error is not observed to reduce with that order. This is mainly because it is a steady case and the value of the time step is fixed.

Following the accuracy assessment, an analysis of computational efficiency is conducted using a well-known test case (see Fig. a), previously simulated and compared against measurements using the SERGHEI framework . The details of the experimental set-up are described in , where the domain dimensions are L=36 m in length, B=3.6 m in width. As the initial condition, an initial water depth of 0.4 m in the reservoir region and 0 m in the right-hand region are imposed, modifying the value of the initial condition in the right-hand region compared to the original experiment to check the accuracy of the model when wet–dry situations occur. The comparative analysis is based on increasing the number of particles to evaluate the model computational cost for each configuration. Moreover, this case is used as a validation of the model's ability to correctly resolve wet–dry situations. Reflective boundary conditions are applied everywhere, maintaining a constant number of particles throughout the simulation. The Runge–Kutta offline method was updated using different time periods (5Δt, 10Δt, 20Δt and 50Δt, where Δt is the hydrodynamic time step) to analyze the influence of this variable in the temporal evolution of the particle position.

The computational costs for the different configurations with different particle numbers are listed in Table , where only the Runge–Kutta offline computational costs updated every five time steps are shown because it is the less computationally efficient case for the simulated Runge–Kutta offline combinations. The computational time for this simulation without particles was 12.15 s, and an overhead ratio relative to the hydrodynamic-only run is computed for each case. The forward Euler method consistently demonstrated shorter computational times compared to other methods across all configurations. In addition, four states during the simulation are depicted in Fig. , illustrating realistic particle trajectories simulated with the Euler scheme without traversing dry cells or climbing over buildings. However, when the Runge–Kutta offline is updated using long time periods, the temporal evolution of particle position can be illogical, as shown in Fig. , where some particles can be seen climbing over buildings. Finally, the differences between the methods' results are shown in Fig. , where the L2-norm per particle is derived from the differences between models, taking the Runge–Kutta online method as the basis for comparison. The error between the Euler and Runge–Kutta methods is lower than the errors between the Runge–Kutta and Runge–Kutta offline methods, indicating that the Euler method provides results more similar to the Runge–Kutta method while maintaining higher computational efficiency. In addition, as the update time for the offline Runge–Kutta method increases, the difference between the online Runge–Kutta and the offline Runge–Kutta increases too, due to the fact that being a transient case, not updating frequently can lead to a high error.

A longitudinal channel with symmetrically shaped cavities was simulated, using a mesh with a resolution of 2×2 cm (see Fig. a). The objective is to check whether a symmetrical distribution of particles is achieved across the upper and lower cavities ; 105 particles were simulated with an initial distribution as shown in Fig. a. Given the best compromise between accuracy and computational efficiency in the last test cases, the Euler method was only employed for this simulation. The temporal evolution of the particle position with and without turbulence was compared (see Fig. ), with the evolution of particle numbers in each pair of upper and lower cavities depicted in Fig. . The presence of turbulence resulted in greater dispersion in particle distribution and more realistic trajectories compared to simulations without turbulence. This is further evidenced in Fig. , where the number of particles within a cavity is higher when turbulence is considered. The symmetry between the upper and lower cavities (see Fig. ), regardless of whether turbulence is added or not, provides a further argument for the accuracy of the Lagrangian model.

An important observation is that with coarser grid resolutions, the velocity field may not be sufficiently resolved to drive particles into the cavities. To explore this, the same simulation was repeated with and without dispersion using a different grid resolution: 20 cm. As shown in Figs.  and , the coarser mesh failed to guide particles into several cavities, with some receiving no particles at all. In contrast, the finer resolution (see Fig. ) captured the expected cavity entry more accurately. This highlights a key limitation of coarse Eulerian velocity fields, which can be partially mitigated by including the random-walk model: it enhances subgrid-scale transport and improves the physical plausibility of particle trajectories. Although this test does not constitute formal model verification, the observed symmetry in particle distributions – particularly under the influence of turbulence – provides soft validation that the Lagrangian model behaves in a consistent and physically reasonable way, without introducing spurious asymmetric artifacts.

The implemented Lagrangian model may have applications in ecohydrology, such as seed, pollutant or nutrient transport. Consequently, a case involving a precipitation event over the Arnás catchment (see Fig. a) was simulated to show the potential application of this model. This catchment is located in the Borau valley (northern Spanish Pyrenees). In the last decades, this region has been an area of exhaustive study of different hydrological processes . Physical processes such as rainfall, infiltration, runoff generation and gully formation have been simulated in detail using various methods . The primary objective of simulating this case was to assess its potential utility in environmental applications. Information derived from conservative tracers and field experiments measuring isotopes in water has been widely used in hydrology to characterize hydrological pathways and connectivity, with ideas such as transit or residence times, and travel time distributions . Here, a minimal example is presented to demonstrate how analogous ideas can be investigated within the proposed computational modeling framework.

The catchment was discretized at 5 m resolution, for a total of 1.1×105 computational cells; 104 particles were randomly placed in the domain as initial condition for the LPT model, and the catchment was initially dry for the shallow water solver. As in the previous test case, the Euler method was exclusively used for the simulation of the Arnás case, based on the optimal balance between accuracy and computational efficiency observed in the initial test cases. Two real precipitation cases (see Fig. b and c) were simulated to analyze the particle trajectory behavior under different hyetographs. In both events, particles are mostly transported through the main gullies. The run times for the simulations of events 1 and 2 were 317.75 and 177.02 s, respectively, corresponding to run-time increase ratios of 2.39× and 1.10× compared to the same simulations without particles.

In addition, the total travel distance per particle, the particle travel time (travel time), and the time during which the particle was at rest (stop time) are shown in Figs.  and for each precipitation event. These figures provide a statistical overview of particle dynamics during the runoff events. Physically, these metrics offer insights into the hydrological processes at play. The travel distance (displayed in Fig. a) reflects the efficiency of the drainage network of the basin to transport the debris, which can be seen in Fig. a and b, where most particles travel a distance very similar to the initial one. Moreover, Fig. a shows that a subset of particles from event 2 travels much farther than the rest (around 10 km). This is due to the properties of this precipitation event, which features less intense but longer rainfall than event 1. This extended duration causes particles to travel for longer periods (with shorter stop times), as also observed in Fig. d. Furthermore, this histogram reveals regions where particles tend to accumulate or remain at rest for extended periods, typically due to reduced flow velocities or topographical barriers, which may indicate zones of storage or potential sediment deposition. However, as shown in Fig. , the set of particles with high travel distances does not influence the last hours of the temporal evolution of the total travel distance (see Fig. ) because in both events the majority of the particles leave the domain. However, the difference between the events is higher at the beginning of the case, having a big difference between both hyetographs in the first hours. In event 1, the longer travel times observed for most particles in Fig. c, where particles tend to remain approximately at rest for extended periods, suggest intermittent flow conditions, possibly due to intermittent rainfall, leading to temporary particle deposition. In contrast, event 2 exhibits more uniform movement and shorter stop times, probably due to less intermittent rainfall that fosters persistent particle transport. Thus, these histograms not only quantify particle behavior but also provide insights into the underlying hydrological and geomorphological processes shaping the catchment's response to precipitation events.