By definition, a Lagrangian description of a system consists of following individual particles along their trajectories as opposed to the classical Eulerian description usually used in Elmer, which focuses on the variation of system variables at fixed locations (the grid points). The SL method is based on a Lagrangian discretization of the transport equation but uses the Eulerian velocity field to determine the particle velocities and, therefore, their trajectories. The velocity and pressure fields are typically solved following Stokes-like equations with Elmer's standard continuous Galerkin formulation with stabilized equal-order linear elements. The resulting velocity field is continuous at element nodes, ensuring smooth trajectories for particle advection. In the absence of source/sink, particles are then simply used to carry advected field from their previous to their current locations. To this end, particles are initialized at the mesh nodes (they can also be initialized at the center of the elements or at integration points) and tracked backward in time following the velocity field u. In the present implementation, particles are not persistently stored throughout the entire simulation but are periodically reinitialized from the Eulerian field at a user-defined interval (see Sect.  for an example of the impact of this choice). When their previous position is recovered, the value of the advected field is reconstructed from the Eulerian mesh using continuous Galerkin (CG) shape functions and assigned to the corresponding nodes at the current time step. These nodal values are subsequently projected to the integration points when required for source-term and velocity-update evaluations (the principle of the method is illustrated in Fig. ). This is done by evaluating the following integral: 2rt-Δt=rt+∫tt-Δtudt where r is the position vector, t is the current time step at which we want to evaluate the new value and new position of the transported variable q, and t-Δt is the previous time step. In order to account for the spatial variability of the velocity field in the reconstruction of the particle trajectory, the integral in Eq. () is evaluated by dividing the main time step of the simulation Δt into a discrete number N of internal time steps δt so that: 3∫tt-Δtudt=∑i=1Nri. with ri the position vector at internal time step i (the red dots in Fig. ). Note that the value of N is a trade-off between an increasing computation time for large N and inaccuracies in the path description, leading to an inaccurate evaluation of the position rt-Δt and therefore the particle value q(rt,t), for small N. The integral may be evaluated using a first-order explicit scheme or a second-order Runge–Kutta scheme. In the first-order scheme, a quadratic correction of the velocity u accounts for the variation of the velocity field across a particle's path, improving the tracking: 4u=u(t,ri)+12∇u(t,ri)⋅u(t,ri)δti with u(t,ri) and ∇u(t,ri) the velocity and velocity gradient at the current time step t and evaluated in ri, respectively. This evaluation is conducted by determining the location of the particle in the element, solving for the local coordinates, and interpolating from nodal values through shape functions. When using a second-order Runge–Kutta temporal discretization, the method already contains inherent quadratic terms, and the correction is not applied. However, the correction of Eq. () is quadratic in the velocity field and can lead to oscillations in the computed field, resulting in non-physical particle positions. When the particles have been advected, the field is evaluated from: 5q=q(rt,t)=q(r,t-Δt) where the variable q in each node depends on the interpolated value of q at the initial position and the earlier time step.

A source term χ that depends on the evolution of the particle along the path integral, following a function χ(q,t), can be evaluated over time. This corresponds to solving: 6It(χ)=∫t-Δttχ(q,t)dt The new value of q is then reevaluated as the combination of Eqs. () and (): 7q=q(rt-Δt,t)+It(χ)

Back-tracked particles have to be localized within the mesh at each internal time step δi to capture the velocity of the particle. The simplest way to find this position is to localize the particle's new location using the coordinate of the particle to retrieve its position into the mesh. The method is accurate but very inefficient for unstructured meshes, making it very greedy in terms of computation time and memory access. In-cell test algorithms have been developed to overcome this issue e.g.. The in-cell test allows to back-track the mesh cell where the particle resides, without the need for additional searches over the entire mesh. The shape functions associated with the nodes within the element are then evaluated, and the interpolation is performed using the nodal values.

Our particle-tracking algorithm is based on the method developed by . It involves tracking the motion of particles from element to element by checking whether or not an element face is crossed over an internal time step. Similarly to , let's define a particle with an initial position r(t) corresponding to a node. For each internal time step δi of the back-tracking procedure, a check is performed to evaluate whether or not one of the faces of the element has been crossed. This is done through geometrical considerations via the calculation of determinants between the vectors formed by the initial and final locations of the particle over the internal time step on the one hand, and the vectors formed by the positions of the nodes of the considered element face on the other hand. If it turns out that a face has been crossed, the face index is used to determine which element, if any, is on the other side of the face. Then, the algorithm is continued from this new element for each remaining internal time step δi until the end of the main simulation time step Δt. While the method requires some complex geometrical tests, it is shown that computational time scales approximately linearly (O(Np)) with the number of particles (Np), as only local operations are performed. This makes the method faster than most tree-based localizations that are shown to scale logarithmically (O(Nplog⁡Np)). More detail on the particle location and element crossing, as well as the interaction with boundary interfaces can be found in the Appendix .

In Elmer/Ice, the advection equation Eq. () can also be solved by applying a discontinuous Galerkin method (DG). Similarly to the SL algorithm, the flow solution does not rely on a DG formulation; only the advection equation employs a DG formulation. The implementation of the method mostly relies on the work of for solving first-order linear hyperbolic equations. In this method, the stability of the numerical scheme (i.e. the stabilization of the oscillations near the discontinuities) is ensured by adding a jump-penalty term to the discretized Eq. () without requiring upwind stabilization or other terms. Only a short summary is presented here.

To apply the DG method with jump-penalty stabilization to the advection equation, we begin by rewriting Eq. ( in its weak form by multiplying it by a discontinuous test function vh and integrating over each element T: 8∫T∂qh∂tvhdx+∫Tu⋅∇qhvhdx=∫TSvhdx. Integrating the advection term by parts, we obtain: 9∫Tu⋅∇qhvhdx=-∫Tqhu⋅∇vhdx+∫∂Tu⋅nqhvhds, where n is the outward normal on ∂T.

The DG formulation can then be expressed as: ∑T∈Th∫T∂qh∂tvhdx-∑T∈Th∫Tqhu⋅∇vhdx+∑e∈Eh∫e{u⋅nqh}[[vh]]ds10+∑e∈Eh∫eαe[[qh]][[vh]]ds=∑T∈Th∫TSvhdx. where the third term ∫e{u⋅nqh}[[vh]]ds captures the advection across the element interface e using the average flux, with [[vh]] being the jump operator representing the difference in vh between the two sides of e, and {u⋅nqh} represents the mean flux across e. The fourth term ensures stability by penalizing large jumps in qh across element edges, where αe is a penalty parameter, often set as αe=|u⋅n|2. This penalization helps prevent spurious oscillations and improves the convergence of the solution.

This classical rotating-disc experiment is designed to assess the ability of the advection schemes to preserve sharp geometrical features and global mass under purely advective motion. Because the analytical solution corresponds to a rigid-body rotation, any deformation or diffusion of the discs can be directly attributed to numerical errors.

This second experiment extends the analysis to a three-dimensional setting and focuses on the influence of time stepping, parallel domain decomposition, and computational cost on tracer advection.

We build a 3D model of the region and simulate the ice flow using the state-of-the-art Stokes flow model Elmer/Ice (see Appendix ), following the initialization procedure described in Appendix . The model is discretized on an unstructured finite-element mesh using stabilized equal-order linear wedge elements for velocity and pressure. Numerical integration is performed using 44 Gauss points per element (11 points on the triangular face and 4 along the vertical direction). Although the velocity and pressure are approximated with first-order Lagrange elements, this relatively high quadrature order is required to accurately integrate the nonlinear viscosity and stabilization terms within each element. Stresses are computed at these integration points and subsequently interpolated to the mesh nodes, where the damage variable is evaluated. Once initialized, we simulate the evolution of damage over 50 years.

Our damage model is based on Continuous Damage Mechanics (CDM) and follows the same physical approach as and : damage is created where the maximum tensile principal stress exceeds a threshold and is advected downstream with the flow. The corresponding advection equation with a source term is solved using either the SL or the DG solver described previously.

To keep the case as simple as possible, we use a constant source term for damage and focus only on the creation of surface damage (i.e. we do not include basal damage due to water pressure in Eq. ). We also ignore feedbacks between damage and viscosity (see Eq. ). Details of the damage model are presented in Appendix . A key parameter of the model is the stress threshold σth at which damage occurs. In previous studies, σth is often taken between 0.1–0.3 MPa e.g.. Here, we select a particularly high value of σth=0.55 MPa to limit damage production (χ>0) to areas clearly identified as crevasse onset regions in observations . Additionally, mountainous regions and steep slopes are excluded from the simulations as these areas are unlikely to sustain ice cover.

Our synthetic experiments indicated that using longer SL advection time steps (Δt) combined with multiple internal time steps (δi) helps limit the interpolation errors that accumulate over repeated advection cycles. This issue is particularly acute in high-resolution ice-flow models (e.g. Δx∼1 km), where small time steps (e.g. Δt∼10-2 year) are typically required for the flow solver to limit feedbacks between vertical velocities and free-surface evolution.

To address this, the SL advection step is executed with a larger time step than the flow solver (Δt>Δtflow), reducing both interpolation frequency and computational cost. Because feedbacks between damage and viscosity are disabled in this experiment, this reduced update frequency does not affect the flow solution. In a fully coupled setup, such decoupling would introduce a trade-off between numerical diffusion and the accuracy of damage-flow interactions. However, as our focus is solely on evaluating numerical diffusion, deactivating the feedback allows us to better compare stepping choices.

We conduct a set of eight simulations of damage evolution using the same steady-state geometry, ice flow, and damage model parameters. Each simulation covers 50 years of damage evolution with SL advection time steps ranging from Δt=10-2 to 5 year. The same simulations are performed using the DG advection scheme for comparison.

The simulations reproduce the main damage structures observed in the Amundsen Sea sector, with damage primarily generated in shear margins near the grounding line of Pine Island Glacier and subsequently advected downstream along flowlines that follow patterns observed in satellite imagery e.g.. Over the Thwaites Ice Shelf, damage is more widespread but still aligns with the general patterns inferred from observations. Both advection schemes capture the large-scale transport of damage from upstream source regions toward the ice shelves.

In a purely advective setting without numerical diffusion, damage generated in localized source regions should remain laterally confined to the downstream flowlines along which it originates. To visualize potential spreading caused by numerical diffusion, we overlay flowlines with spacing ranging from about one to three elements (approximately 500 m to 5 km depending on the region).

For the SL simulations, decreasing the advection time step Δt–and therefore increasing the number of particle–mesh interpolation operations–leads to increased smoothing of the vertically integrated damage field, producing progressively smoother damage patterns (Fig. ). This effect is particularly noticeable over Thwaites Glacier and its ice shelf, where margins and sharp structures become increasingly diffuse as Δt decreases. For instance, for Δt=1 yr (Fig. a), only limited lateral spreading is observed: damage generated upstream remains largely confined within one flowline spacing (Dmean<10-3). Smaller time steps of Δt=10-1 yr (Fig. b) and Δt=10-2 yr (Fig. c) lead to progressively stronger lateral spreading, with damage originating from neighboring ice streams occasionally merging downstream. In addition, we observe cumulative smoothing and a gradual loss of damage intensity along the flowlines as the signal is advected farther from its source, an effect that is particularly visible along the shear margins of Pine Island Glacier and consistent with the non-conservative nature of the particle–mesh remapping.

The DG simulations exhibit a different spreading structure. While the large-scale damage patterns remain comparable, diffusion occurs preferentially along the flow direction. As a result, damage features tend to extend farther downstream from the source regions for larger time steps, producing elongated streaks aligned with the velocity field. This contrasts with the SL solution, where smoothing arises primarily from particle–mesh interpolation and therefore appears more isotropic, with only a secondary influence from mesh orientation. At the same time, the DG formulation preserves tracer intensity more effectively, reflecting its locally conservative nature. These differences are consistent with the behavior observed in the synthetic benchmark experiments. Small oscillatory patterns can also be observed along some DG damage trajectories, appearing as triangular structures aligned with the mesh elements. These features likely reflect element-wise reconstruction and limiting operations used in the DG discretization. Although these mesh-scale artefacts remain localized and do not significantly affect the large-scale damage distribution, they highlight differences in how the two schemes handle steep tracer gradients during advection.

We can also assess numerical diffusion by examining three vertical damage profiles at different locations (see locations in Fig.  and profiles in Fig. ). On Profile 1 (Fig. )a, d), located within a shear margin of Pine Island Glacier and downstream of a damage source, the overall damage distribution remains relatively similar for both advection schemes. However, for the SL simulations, decreasing Δt leads to a reduction in the maximum damage near the surface. The DG simulations show the opposite tendency, with the surface maximum decreasing as Δt increases. We also observe a deeper penetration of damage into the ice column as Δt decreases or increases, depending on the method but this effect is stronger with SL. One exception occurs for very large SL time steps (e.g. Δt=5 yr), which produce slightly lower damage within the upper ∼100 m of the ice column. This behavior may result from reduced numerical diffusion of damage originating from adjacent flowlines. Profile 2, located downstream of a damage source of Thwaites Glacier, exhibits a similar pattern (Fig. b and e). Conversely, Profile 3 which is situated between two damage sources and is not downstream of any specific source, exhibits low to no damage for Δt=5 yr SL simulation. In the absence of numerical diffusion, the ice should remain undamaged at this location. However, due to numerical diffusion, damage appears and increases rapidly as Δt decreases for the SL, before stabilizing once the particle displacement per time step becomes small enough that further interpolations have a lesser impact on the results (Fig. c). The DG simulations also show the effect of numerical diffusion but to a lesser extent. Overall, the vertical profiles for both methods tend to converge toward similar damage distributions as Δt decreases for SL and increases for DG, suggesting that numerical diffusion approaches a plateau for both schemes

In addition to the differences in numerical diffusion, the two advection schemes exhibit markedly different computational costs in the present configuration. In the Antarctic experiments, the DG simulations require an advection time step of order Δt≈10-2 yr. In contrast, accurate SL solutions are obtained with Δt=1 yr using N=10 internal timesteps for trajectory integration. Under these conditions, the SL advection step is roughly ∼50–100 times cheaper than the DG solver at the same mesh resolution. While the SL still shows diffusion at Δt=1 yr, this computing time difference could allow for the SL solver to run with ∼50–100 times more particles for a similar cost as the DG solver at Δt≈10-2 yr.