Ice is a highly viscous, shear-thinning fluid, with conservation of momentum described by the Stokes equations, 1∇⋅σ=ρig, where σ is the Cauchy stress tensor, ρi is the ice density, and g=(0,0,g) is the gravitational acceleration. The Cauchy stress tensor can be split into pressure, p, and a deviatoric stress tensor, τ, 2σ=τ-pI, where I is the identity matrix. The conservation of mass is expressed as an incompressibility condition, 3∇⋅u=0, where u=(ux,uy,uz) is the velocity vector. In order to relate the nonlinear response of the strain rate of ice to an applied stress, Glen's flow law is used, 4τ=2η(ε˙)ε˙, where ε˙ is the symmetric part of the velocity gradient tensor, i.e. ε˙=1/2(∇u+∇uT). The effective viscosity, η=η(ε˙), is defined as 5η=12A-1/nε˙crit2+12||ε˙||F2(1-n)/2n. Here, the effective strain-rate, ||ε˙||F, is the Frobenius norm of the strain rate tensor, which can be written ||ε˙||F=tr(1/2ε˙2). The rheological parameter, A, controls the fluidity of the ice and is spatially constant in this study, but is temporally-varying in specified simulations (Sect. ). A regularisation term, ε˙crit2=10-15a-1, is included to numerically stabilise the viscosity as the strain rate approaches zero.

The Stokes equations are solved on a domain Ω, with a Cartesian coordinate system, x=(x,y,z)∈R3. Although the equations show no explicit dependence on time, time dependence enters the problem because the domain, Ω, changes in time. Boundary conditions are applied at the domain boundary ∂Ω (Fig. ). The ice-atmosphere interface is denoted by Γa, the ice-bed interface by Γb, the ice–ocean interface by Γo, the calving front by Γc, and the ice divide by Γd so that ∂Ω=Γa∪Γb∪Γo∪Γc∪Γd. During a transient simulation, the floating and grounded boundaries evolve to reflect changes in the grounding-line position. At the ice-atmosphere interface, Γa, a stress-free boundary condition is applied, 6σ⋅n^=0, where n^ is the normal vector perpendicular to the ice surface and pointing outwards and is implicitly included in the weak form.

The ice-bed interface is subject to a Weertman-type friction law motivated by the set up in the Marine Ice Sheet Model Intercomparison Project MISMIP,, 7ti⋅(σ⋅n^)=-C|ub|m-1u⋅ti, where for each i∈{1,2}, ti represents a vector tangential to the plane defined by the ice-bed interface. An impenetrability condition is applied to ice that is in contact with the bed, 8u⋅n^=0.

At the boundaries, which are in contact with the ocean, Γo and Γc, the stress normal to the boundary is prescribed by the ocean pressure as, 9σ⋅n^=-pwn^, where n^ is the outward-pointing vector normal to the surface and the ocean pressure, pw(z,t), is defined by 10pw(z,t)=ρwg(zsl(t)-z) where ρw is the ocean density and zsl is the sea level. Where the ice–ocean interface, Γo, and the ice-bed interface, Γb, meet, the respective boundary conditions are discontinuous at the nodes representing the grounding line . Finally, at the ice divide in the 2D MISMIP simulations, Γd, a symmetry condition is applied, 11u⋅n^=0.

In the 3D Ekström Ice Shelf experiments, the impenetrability condition of Eq. () is replaced with a normal velocity Robin boundary condition with a coefficient chosen such that through-flow is negligible. This boundary condition takes the form 12n^⋅(σ⋅n^)=-Cnu⋅n^. This formulation alleviated spurious artifacts at the ice divide, which occurred when implementing a strong-form impenetrability condition . We choose a value of Cn=1012MPam-1a in this weakly enforced boundary conditions.

The Stokes equations are written in their weak form in order for the velocity and pressure to be solved using the finite element method with Elmer/Ice . Multiplying by a test function, v:Ω→R3, and integrating over the model domain, Ω, the weak form of the Stokes equations reads 13∫Ω(∇⋅σ)⋅vdΩ=∫Ωρig⋅vdΩ. In weak form, the incompressibility condition is written 14-∫Ω(∇⋅u)qdΩ=0, where q:Ω→R is also a test function. Inserting the boundary conditions above results in the weak form of the Stokes equations applied to a marine-terminating ice sheet 15-∫Ωτ:∇vdΩ-∫Γo∪Γcpwn^⋅vdΓ-∫ΓbC|ub|m-1ub⋅vdΓ+∫Ωp∇⋅vdΩ=∫Ωρig⋅vdΩ. The friction law is nonlinear, so it must be treated carefully when solved numerically. The nonlinear coefficient, C|ub|m-1, is evaluated at the previous nonlinear iteration, j-1, for inclusion in the current iteration, j. To ensure that C|ub|m-1→∞, including during initialisation when |ub|=0, the coefficient is capped at 1020.

Our simulations assume a fixed calving front, but we envisage that a dynamic calving front could be numerically stabilized using a similar approach to FSSA by treating relevant terms implicitly.

The upper ice surface, z=zs(x,y), evolves subject to an evolution equation defined by, 16∂zs∂t+ux∂zs∂x+uy∂zs∂y=uz+as, where as=as(x,y) is the surface accumulation rate. Similarly, the lower ice surface evolves subject to 17∂zb∂t+ux∂zb∂x+uy∂zb∂y=uz+ab, where ab is the basal melt (if ab>0) or accretion rate at the lower ice surface. The modelled ice is subject to constraints to ensure that the ice thickness does not become too thin. A constraint is placed on the upper ice surface so that 18zs(x,y,t)-Hmin⁡≥zb(x,y,t), The lower ice surface is subject to a constraint ensuring impenetrability, 19zb(x,y,t)≥b(x,y). Here, b(x,y) is the bed elevation and Hmin⁡ is the minimum ice thickness. The elevation of the lower ice surface, zb, also depends on the dynamics of the grounding line, which is determined by the balance between the contact force and the water pressure.

In order to solve the free-surface equations, they are discretised in time as 20zsk+1+Δtux∂zsk+1∂x+Δtuy∂zsk+1∂y=zsk+Δt(uz+as), and 21zbk+1+Δtux∂zbk+1∂x+Δtuy∂zbk+1∂y=zbk+Δt(uz+ab). In weak form, the equations are multiplied by a test function, ϕ, and integrated over the upper and lower ice surfaces, so that 22∫Γazsk+1+Δtux∂zs∂x+Δtuy∂zs∂yϕdΓ=∫Γa[zsk+Δt(uz+as)]ϕdΓ and 23∫Γazbk+1+Δtux∂zb∂x+Δtuy∂zb∂yϕdΓ=∫Γa[zbk+Δt(uz+ab)]ϕdΓ. A backwards difference formula (BDF) time-stepping scheme of first order (i.e., implicit Euler) is employed.

To stabilise the contact problem, a numerical stabilisation scheme called sea spring was implemented on the lower ice surface in Fig. . The sea spring numerical stabilisation scheme predicts the lower ice surface, zb, in the next time step and adjusts the ocean pressure accordingly. The ocean pressure term before numerical stabilisation reads 24σnn|b(x,y,t)=-ρwg(zsl-z) and is evaluated on the current time step, t. Here, zsl represents the sea level. The lower ice surface elevation at the next time step is predicted using an estimate of the vertical displacement, so that 25zb(x,y,t+Δt)≈zb(x,y,t)+(u⋅n^)Δt×1+∂zb(x,y,t)∂x2+∂zb(x,y,t)∂y2, where Δt is the time step size and t+Δt is the next time step. The ocean pressure term therefore becomes 26σnn|b(x,y,t)=-ρwg{zsl-zb(x,y,t)+(u⋅n^)Δt×1+∂zb(x,y,t)∂x2+∂zb(x,y,t)∂y2}

The Free-Surface Stabilisation Algorithm (FSSA), was introduced by to a Stokes problem with a free surface applied to mantle convection simulations. This method was later extended to Stokes free-surface problems in grounded ice-sheet modelling, proving successful in increasing the largest stable time step size . Here we will derive the FSSA for floating ice (Fig. ). The FSSA predicts the free-surface elevation at the next time step, t+Δt, and includes extra terms as boundary conditions to account for the adjustment in ice load. However, contrary to the expression for the ocean pressure, the free surface does not enter explicitly in the formulation for the ice load, but rather through the integral associated with the weak form. The FSSA is derived from Reynold's Transport Theorem, 27ddt∫Ω(t)fdΩ=∫Ω(t)∂f∂tdΩ+∫∂Ω(t)(u⋅n^)fdΓ, where f is a physical quantity, Ω(t) is the model domain at time step t and ∂Ω(t) is the domain boundary at time step t. For our purposes, we take f=-ρig⋅v, i.e. the right hand side of the Stokes equations multiplied by the test function, v. Reynolds Transport Theorem enables the Stokes equations to account for a spatially evolving domain by incorporating the effect of gravitational forces at the next time step, thereby treating the right hand side implicitly. Assuming that f is time independent, and discretising in time, an approximation is found for the integral of f over the time-discretised domain, at the next time step, Ωk+1, based on the current time step, k, so that 28∫Ωk+1fdΩ≈∫ΩkfdΩ+Δt∫∂Ωk(u⋅n^)fdΓ.

The driving force term, i.e. the right hand side of the Stokes equations, therefore becomes 29∫Ωk+1ρig⋅vdΩ≈∫Ωkρig⋅vdΩ+Δt∫Γak(u⋅n^)ρig⋅vdΓ+Δt∫Γok(u⋅n^)ρig⋅vdΓ, where the last two terms represent the predicted upper and lower ice surfaces at the next time step, with an impenetrability condition, u⋅n^=0, at the ice divide and at the ice-bed interface. FSSA is not applied at the calving front because the mesh does not move horizontally here. If an accumulation and melt rate are to be included, two additional terms are added to represent the resulting vertical displacement, 30∫Ωk+1ρig⋅vdΩ≈∫Ωkρig⋅vdΩ+Δt∫Γak(u⋅n^)(ρig⋅v)dΓ+Δt∫Γok(u⋅n^)(ρig⋅v)dΓ+Δt∫Γak(ase^z⋅n^)(ρig⋅v)dΓ+Δt∫Γok(abe^z⋅n^)(ρig⋅v)dΓ

The Stokes equations weak form (Eq. ) with the FSSA and the sea spring numerical stabilisation scheme therefore become 31-∫Ωkτ:∇vdΩ-∫Γok∪ΓckρwgΔt(ũk+1⋅n^)×1+∂zb∂x2+∂zb∂y2n^⋅vdΓ-∫ΓbkC|ũbk+1|m-1ũbk+1⋅vdΓ+∫Ωkp∇⋅vdΩ-θΔt∫Γak(ũk+1⋅n^)(ρig⋅v)dΓ-θΔt∫Γok(ũk+1⋅n^)(ρig⋅v)dΓ=∫Ωkρig⋅vdΩ-∫Γok∪Γckρwgzn^⋅vdΓ+θΔt∫Γak(ase^z⋅n^)(ρig⋅v)dΓ+θΔt∫Γok(abe^z⋅n^)(ρig⋅v)dΓ where we have assumed the sea level to be zsl=0 and the ocean pressure was applied only on the portion of the calving front in contact with the ocean, Γck. Here, ũk+1 is an approximation of the velocity at the next time step, k+1. Since the calving front, Γo, does not move horizontally, the FSSA terms are identical to zero there. Here, the parameter θ has been introduced to allow adjustment of the FSSA terms. A value of θ=0 disables the FSSA terms, while a value of θ=1 fully activates the FSSA terms. Finally, the weak form of the incompressibility condition reads 32-∫Ωk∇⋅ũk+1qdΩ=0.

Note that whether the FSSA and sea spring are active or not at the lower and upper ice surfaces has an effect on the force balance at those surfaces due to the opposing ice load and ocean pressure. When both the sea spring numerical stabilisation scheme and FSSA are active, both the ocean pressure and the ice load are evaluated at the next time step, which results in a balance of forces. When only the sea spring numerical stabilisation is activated, the ocean pressure is evaluated at the predicted surface at the next time step. As a result, the ocean pressure contribution in the Stokes equations is higher in an advancing case, i.e. when the ice is thickening, compared to evaluating the ocean pressure at the current ice surface. The opposite is true in the retreating case, i.e. the ocean pressure contribution is lower with sea spring because it predicts thinner ice. In contrast, including the FSSA term at the lower ice surface predicts a higher ice load in an advancing case due to the prediction of thicker ice and a lower ice load in a retreating case due to the prediction of thinner ice.

During a transient simulation, the grounding line position must be allowed to migrate to reflect the altering ice geometry and velocity. Once a new grounding line position is determined, the application of boundary conditions must be altered accordingly. To determine whether to apply a friction law and an impenetrability condition or whether to apply ocean pressure and a free-slip condition at a particular node, the normal stress and the ocean pressure are examined through an algorithm presented in .

During discretization and global assembly of the Stokes equations, the left hand side is split into the mass matrix, K, and the solution vector, U and the right hand side is the force vector, F. Once solved, the residual can be obtained, 33R=KU-F. When an exact solution is obtained, only the components in R corresponding to nodes where a Dirichlet boundary condition is applied are non zero and are equal to the ice load force, or equivalently the normal stress, at that node. This nodal force is used to evaluate whether the ice-bed boundary conditions should be applied, i.e. if zb(x,y,t)=b(x,y) and -σnn|b>pw(zb,t), or whether ice–ocean boundary conditions should be applied, i.e. zb(x,y,t)>b(x) or zb(x,y,t)=b(x) and -σnn|b≤pw(zb,y).

When solving the Stokes equations, the boundary conditions are first determined a priori by checking only whether zb(x,y,t)=b(x,y) and applying ice-bed boundary conditions or otherwise applying ice–ocean boundary conditions. Once the Stokes equations are solved to a convergence tolerance of 10-3, the balance of forces is re-evaluated and the boundary conditions are altered once in the nonlinear Stokes iterations per time step.

Once the Stokes solver has converged, the resulting velocities are used to calculate the free-surface evolution. Here, we describe the evolution of the lower ice surface. Again, a system of equations in weak form must be solved, 34Mzbk+1=f, where zb is the solution vector for the ice surface elevation at the next time step, M is the matrix of coefficients and f is the vector representing all terms in Eq. () that do not contain the variable zbk+1. While solving the free-surface evolution equation, a node zb,j=zb(x,y,t) is marked as active if it does not satisfy the condition b(x,y)≤zb(x,y,t)<zs(x,y,t).

For each node zb,j marked as active, the system matrix is altered so that Mij=δij, where δij is the Kronecker delta defined by δij=1 if i=j and δij=0 if i≠j. If the node is marked as active because zb,j<bj, the corresponding component of f is set to fi=bj. Alternatively, if zb,j≥zs,j, the corresponding right-hand side component is set to fi=zs,j.

The system is solved repeatedly until the convergence tolerance is reached. Between each solve, the residual is solved, 35r=Mzb-f. This is used to determine the set of active and inactive nodes in the next solve.

If a node was previously active because zb,j<bj has a residual rj<0, then it is marked inactive. On the other hand, if a node was previously active because zb,j≥zs,j, then a residual of rj>0 results in the node being marked inactive. Similarly, a node can become active if it satisfies b(x,y)≤zb(x,y,t)<zs(x,y,t).

Simulations with FSSA activated (θ=1) are performed with time-step sizes of Δt=0.5,10 and 50 years and varying grounding zone resolution (Table ). A reference simulation of the full advance and retreat phases without the FSSA terms (θ=0) is performed for a time step size of Δt=0.5 and a grounding-zone horizontal resolution of 200 m (Table ). A finer spatial resolution is computationally infeasible because a time step size of Δt=0.5 is too small to allow for reasonable simulation times. All simulations are subject to a small time step of 0.5 years for 1000 years after the ice fluidity jump in the retreating phase of Exp. 3a. When a small time step size is not used for a short period of time during the ice fluidity jump, the upper and lower surfaces, as well as the velocity field, exhibited oscillatory behaviour, leading to numerical stability.

The MISMIP Exp. 3a is defined by a 2D modelling domain of a marine ice sheet with the x-axis denoting the horizontal direction and the z-axis denoting the vertical direction. We apply the Stokes model and assume invariance in the y-direction. The bed elevation for Exp. 3a is defined as 36b(x)=729-2184.8x7500002+1031.72x7500004-151.72x7500006.

The model domain is initialised with a 10 m thick ice sheet and ice shelf. Initially, the floating ice is prescribed to be in hydrostatic equilibrium, a constraint which is only enforced to construct the initial geometry. The model parameters remain fixed throughout the simulations, with the exception of the ice fluidity A, and are defined in Table . The ice fluidity, A, is altered at 12 time points during the simulation, according to the values provided in Table . The step-wise change in fluidity induces grounding-line advance as the ice becomes stiffer (decreasing A) and grounding-line retreat as the ice becomes softer (increasing A). The experiment is carried out over a period of 285 000 years and demonstrates the typical hysteretic behaviour of the grounding line in areas of alternating pro- and retrograde slopes. As outlined in the MISMIP study , the first 10 000 years of Exp. 3a are prone to numerical instability due to the thin, unrealistic nature of the domain. We chose a numerically-stable set up, which allowed quick simulation of these first 10 000 years. The chosen setup was performed with a time step size of Δt=10 years and a horizontal resolution of Δx=2000m throughout the domain. Each simulation in our experiment, as outlined in Table , is performed beginning with the geometry at t=10000 years.

Meshes are constructed from a 1D footprint mesh, which is vertically extruded to 20 layers, forming a terrain-following mesh consisting of quadrilateral elements. The footprint is refined in the areas either side of the grounding line position, in a 100 km wide area that we refer to as the grounding zone hereafter. The grounding zone resolutions tested are Δx=25,50,100 and 200 m (Table ). Outside the refined area, a gradually decreasing horizontal resolution is applied to avoid large jumps in element size from the refined to the more coarsely resolved area, i.e. to ensure good mesh quality. Every 500 years, the mesh is updated, ensuring that the grounding line always remains within the area of refined horizontal resolution. This is done by finding the grounding line position, xGL, and defining the region within which the horizontal mesh should be refined, i.e. [xGL-50,xGL+50]km. Outside this region, a linearly decreasing resolution is prescribed. The number of nodes in the area with a coarser horizontal resolution is defined in such a way that the number remains the same throughout the simulations. Further implementation details using the Gmsh software can be found in . The simulations with a refined horizontal resolution of Δx=25m contain 98 679 nodes, those with Δx=50m contain 56 679 nodes, those with Δx=100m contain 35 679 nodes and those with Δx=200m contain 25 179 nodes. Each node has 3 degrees of freedom, corresponding to the horizontal velocities, (ux,uz), and pressure, p.

Linear finite elements for velocity and pressure are used, with the Bubbles stabilisation . The choice of linear elements is justified by the convergence tests in , which showed no improvement in accuracy when using quadratic elements for velocity and linear elements for pressure. The Stokes equations are solved using the Multifrontal Massively Parallel sparse direct Solver MUMPS,. In each time-step, first the Stokes equations are solved, followed by the upper free-surface and finally the lower free-surface equation.

Picard fixed point iterations are used to resolve the non-linear viscosity, with a nonlinear system convergence tolerance of 10-5 using a relaxation factor of 0.9. The free surface Eqs. () and () are solved semi-implicitly, using a velocity from the last Stokes iteration (i.e explicit treatment) but an implicit treatment of the surface itself. The linear convergence tolerance is set to 10-6 for both the upper and lower ice surfaces using the biconjugate gradient stabilised (BiCGStab) method, preconditioned using incomplete LU factorisation (ILU0). The minimum ice thickness is set to 10 m and is activated when solving the free-surface evolution equation for the upper ice surface. Due to the variational inequality, which sets the upper and lower limits of the free surfaces, nonlinear iterations with a convergence tolerance is set to 10-5 are performed to solve the free surface evolution equation. The streamline upwind/Petrov-Galerkin (SUPG) method is used for numerical stabilisation of the free surface solvers . The MISMIP simulations were performed in serial on the Dardel high-performance computing system (PDC Center for High Performance Computing), using the CPU partition with AMD EPYC Zen2 processors.

The grounding line first advances and then retreats. Directly after each jump in ice fluidity, A, the grounding line advanced or retreated more rapidly for a period of time before slowing down. The slope of the bed also had an influence on how rapidly the grounding line advanced or retreated. During advance, the grounding line was first on a prograde slope. Once it had advanced enough to reach the area with a retrograde slope, it advanced more rapidly. In the retreating phase, the grounding line retreated more rapidly when on the retrograde slope and more slowly on the prograde slope.

Even for small time-step sizes, much lower than those presented here, no simulation was possible without sea spring. Simulations without the FSSA (θ=0) were not numerically stable with time step sizes of 10 and 50 years. The FSSA-stabilised simulations with a time-step size of Δt=10 and Δt=50 years were numerically stable. The simulations with Δt=10 also showed high accuracy (Fig. ). With this time-step size, the maximum grounding line extent ranged from 1413 km in the case of the simulation with a horizontal grounding-zone resolution of Δx=25m to 1406 km in the case of the simulation with a horizontal grounding-zone resolution of Δx=200m, both close to the maximum grounding line extent of 1414 km in the reference simulation with Δt=0.5, Δx=200 and θ=0.

The simulations with Δt=10 lagged behind the reference simulations during rapid grounding line advance, i.e. when the grounding line position was on the retrograde slope and the grounding line position was unstable. The maximum lag of the grounding line position ranged from of 244 km in the case of the simulation with Δx=25-297m with a horizontal grounding-zone resolution of Δx=200m. During rapid grounding line retreat, when the grounding line position was unstable, the simulations lagged further behind the reference simulation, with the Δx=25 simulation lagging a maximum of 431 km and the Δx=200 simulation lagging a maximum of 494 km behind the reference simulation. There is no clear relation between spatial resolution and the lag. This lag occurs despite the precaution of applying a small time step size of 0.5 years immediately after each ice fluidity jump.

In the simulations with a time-step size of Δt=50 years, the grounding line did not follow the full hysteretic behaviour expected of this experiment (Fig. ), although approximately the same steady state was eventually reached, and the simulation was very accurate in phases of slow migration. The grounding line reached maximum extents ranging between 958 and 962 km (depending on the resolution Δx=25,50,100, and 200 m) compared with a maximum extent of 1414 km in the reference simulation.

In Fig.  there is no significant difference between simulations with and without FSSA for a small time-step size of Δt=0.5 when analysed on the timescale of the entire MISMIP experiment. However, during periods of rapid advance and retreat, a small difference is observed due to evaluating the ice load and the ocean pressure at the same time step (θ=1) compared to at different time steps (θ=0), see Fig. . There is a small influence on the grounding line position, with FSSA simulations having a grounding line further oceanward throughout the simulation time, with the simulation that includes the FSSA being on average 1030 m further oceanward in the advancing phase and 635 m further oceanward in the retreating phase. In other words, the grounding line advances more quickly and retreats more slowly in the simulation with the FSSA active (θ=1) compared to the simulation without the FSSA terms (θ=0). The reasons for this mismatch are discussed in Sect. .

Incorporating FSSA resulted in a significant improvement in computational efficiency compared to simulations without it (Fig. ). Without the FSSA, a time step size of Δt=5 did not converge after a short time and a time step size of Δt=2 converged well in the advancing phase but did not converge in the retreating phase even with the standard short periods of a smaller time step immediately after the viscosity jumps. In contrast, a time step size as large as 250 years is numerically stable using FSSA, albeit with error. The time step size of Δt=50 resulted in a speedup range of 38–43 compared to a time-step of 2, when measured from t=10000 for 500 years and the time step size of Δt=10 resulted in a speedup of 11. In simulations with a horizontal grounding-zone resolution of Δx=25m, the speedup between time step sizes of Δt=10 and Δt=50 was 4.4. Simulations with a horizontal grounding-zone resolutions of Δx=200m showed a similar speed up of 3.8 when comparing time step sizes of Δt=10 and Δt=50.

In order to evaluate our FSSA modelling framework in the context of a 3D, real-world, grounding-line problem, we apply the FSSA-stabilised model to the Ekström Ice Shelf embayment in Dronning Maud Land, East Antarctica (Fig. ). Ekström Ice Shelf has a length of ∼120km compared to a width of only ∼50km and is strongly influenced by lateral buttressing caused by the Søråsen and Halvfarryggen Ice Rises, resulting in a relatively thick ice shelf. Ekström Ice Shelf has been the subject of numerous studies, including comprehensive mapping of the ice shelf cavity and related studies e.g.. Moving to a 3D, real-world setting allows validation of the FSSA across a variety of flow regimes and grounding line behaviours.

In order to model Ekström Ice Shelf, a number of changes were made in comparison to the idealised experiments above. Several parameters are given more accurate values e.g., as laid out in Table . The spatially constant isothermal ice fluidity, A, and the friction coefficient, C, were tuned to values that caused minimal deviation from observed surface velocities in an ad hoc manner given that the purpose of the study is to investigate the applicability of FSSA and not to make projections. The geometry of the initial upper ice surface, the lower ice surface and the bed elevation were derived from . For the surface accumulation rate, as, we used the results of the Regional Atmospheric Climate Model RACMO2.3p1,. The basal melt rates are derived from and a condition is applied to ensure that the melt occurs only at floating nodes. The surface accumulation rate and the basal melt rate are included in the free-surface evolution equations (Eqs.  and ), and the FSSA accumulation terms, i.e. the last two terms of Eq. (). No smoothing of the initial upper and lower ice surface, the bed elevation nor the surface accumulation rate was made. However, significant fluctuations in the basal mass balance caused excess thinning of the ice in isolated areas and numerical instabilities regardless of whether the FSSA was active or not. Therefore, anomalies>4ma-1 were removed and smoothing of the basal melt rate dataset was performed with a Gaussian filter.

The horizontal boundaries of the model domain were chosen along divides and streamlines, and close to the calving front. In the interior portion of the embayment, the low ice thickness (<10m in some areas) and the roughness of the bed caused numerical instability. We therefore chose to reduce the inland extent of the domain slightly to avoid these numerical instabilities, which is justified because the velocities are low in this area and have only a negligible influence on the overall ice-flow dynamics of the downstream embayment.

The 3D finite element mesh is constructed using an unstructured mesh footprint with a horizontal resolution of Δx=500m in an area 3000 m either side of the grounding line and a horizontal resolution of Δx=2000m elsewhere. The horizontal mesh footprint is vertically extruded for a total of 6 layers, allowing for high horizontal resolution at the grounding line and a reasonable compute time. The mesh contains 18 343 nodes, meaning that there are 73 372 degrees of freedom and the simulations are performed with 100 partitions using CPU partitions on the Dardel high-performance computing system with AMD EPYC Zen2 processors. All other details described in Sect.  hold for the simulations of Ekström Ice Shelf.

Simulations varying in time-step size are performed of the Ekström Ice Shelf embayment to investigate the applicability of the FSSA to 3D, real-world, marine-terminating ice-sheet simulations. The time-step sizes chosen are Δt=0.5,5,10 and 20, which were performed with (θ=1) and without (θ=0) the FSSA. The solver set up for the Stokes and free-surface solvers is the same as in the MISMIP experiments. Each simulation is performed for a period of 100 years after which the results are compared. In each time step, the Stokes solver is executed first, followed by the upper and lower free-surface solvers.

Our analysis concentrates on a vertical cross-section, A-A′, of the 3D model domain, which captures the grounded area, grounding zone and floating ice shelf (Figs.  and ). Along this cross-section, the simulated ice thickness reaches a minimum of 519 m and a maximum of 870 m in the grounded ice. In contrast, the minimum and maximum ice thickness in the ice shelf along this cross-section are 178 and 823 m. The grounded ice has a mean thickness of 724 m and the floating ice has a mean thickness of 483 m along this cross-section. Along the cross-section, the basal melt rate, ab, has significantly higher values than the surface accumulation rate, as, with values reaching 1.9 ma-1 compared to 0.4 ma-1, respectively.

The three simulations that include FSSA with time step sizes of Δt=0.5, 5 and 10 years converge in a small number of iterations and result in geometries that are very similar. The FSSA simulation (θ=1) with a time step size of Δt=20 years does not converge within 100 nonlinear iterations of the Stokes solver, even in the first time step. In contrast, the only simulation without the FSSA (θ=0) to converge adequately throughout the simulation of 100 years is the simulation with a time step size of Δt=0.5. In the simulation with a time step size of Δt=5 and without the FSSA (θ=0), the Stokes solver and the free surface solvers converge for the first 7 time steps before numerical instability occurs. In the simulations with the FSSA activated and time-step sizes of 0.5, 5, and 10, the maximum horizontal velocities reach magnitudes of 196, 105, and 74 ma-1. In the short simulation time, the simulations with parameter combinations of Δt=0.5 and θ=0 or θ=1 result in a retreat of the grounding line in the transect A-A′ of 1225 and 770 m, respectively. The simulations with θ=1 and Δt=5 or 10 years show an advance of 8850 m and a retreat of Δt=10m, respectively. A dip in the bedrock causing a cavity in some simulations likely explains the difference in grounding line position.

The simulations with a time step size of Δt=0.5, with and without FSSA, show very little difference in Central Processing Unit (CPU) runtime when run over 100 simulation years, with just a 0.6% difference (Fig. ). However, when increasing the time step size to Δt=5 or Δt=10, the runtimes are 12 % and 7 % of the runtime of the Δt=0.5 with FSSA inactive (θ=0).

The findings of this study demonstrate that inclusion of the Free-Surface Stabilisation Algorithm (FSSA) at the ice–ocean interface significantly increases the largest numerically stable time step size of marine ice sheet simulations. In an idealised, 2D model setup, our results show that a time-step size of Δt=50 years is generally numerically stable but only accurate in phases of slow grounding line advance and retreat and that a time step size of Δt=10 years provides high accuracy throughout most of the simulations. This indicates that the dynamical changes of grounding line problems does not require the small time-steps traditionally used, and that the requirement of such small time-steps is rather due to suboptimal numerical methods with poor stability properties. However, high time-step sizes require a small time step immediately after a viscosity jump during the grounding-line retreat phase. In 100 year, 3D simulations of an embayment, a time step size of 10 years is found to be numerically stable and accurate when including the FSSA. We expect that an even higher time step size may be numerically stable after allowing further relaxation of the simulation, or correction of bed elevation datasets which are prone to inaccuracy in the grounding zones . In light of our results, we recommend the use of an adaptive time-stepping scheme when incorporating the FSSA into Stokes simulations, with a large time step size being used when little change in the velocity and geometry occurs and a small time step size when significant change in the system occurs to ensure accuracy.

The FSSA accomplishes a significant speedup in CPU runtime by predicting where the ice surface will be in the next time step, thereby mimicking an implicit time-stepping scheme when calculating the velocities using the Stokes solver. This allows anticipation and dampening of large changes in the solution, therefore allowing larger time steps to be taken. Furthermore, this work builds on an additional numerical stabilisation scheme applied to the ice–ocean interface, namely the sea spring numerical stabilisation scheme . The sea spring numerical stabilisation scheme is paramount to the stability of such marine ice sheet simulations, with a time step size much lower than those used here not converging without it. The inclusion of both numerical stabilisation schemes results in the ice load and the ocean pressure being evaluated at the next time step, ensuring that a mismatch in forces at the ice–ocean interface no longer occurs. This is evident in Fig. , where in both the advancing and retreating phases, the simulation with the FSSA terms is further oceanward. We hypothesise that this difference during the grounding-line advance is because the simulation with the FSSA terms predicts thicker ice and therefore a higher flux, which results in faster grounding-line advance, which is in line with the theory of . In the retreating case, the opposite holds. The simulation with the FSSA terms predicts thinner ice and therefore a lower ice flux and slower grounding line retreat.

The FSSA is an easy-to-implement approach to improve numerical stability. Since the Stokes model captures the underlying physics more accurately than simplified models, this approach leads to improved overall precision in the results. Further possibilities for a broader application of full Stokes models with the inclusion of FSSA include coupling frameworks with lower-order models . While FSSA enables greater numerical stability in physically complex areas such as grounding-zones, a coupling framework to a lower order model in areas where flow is less complicated will further improve model efficiency. The increased efficiency that FSSA offers has the potential to improve the applicability of simulations of ice stratigraphy, where high resolution and accurate ice flow physics are important . Additionally, the improved efficiency has the potential for full Stokes models to be used in machine learning methods where a significant number of simulations need to be performed . Integrating the modelling framework with FSSA and an adaptive time-stepping scheme will allow the automated increase in efficiency while maintaining accuracy, particularly during rapid change such as grounding-line migration. While the FSSA mimicks an implicit time-stepping scheme, a fully implicit scheme would offer additional model accuracy and efficiency .

A number of problem-specific challenges remain, including the treatment of the minimum thickness constraint in combination with the FSSA, which posed a problem in initial simulations of the Ekström Ice Shelf embayment. In addition, further analysis of the discretisation schemes for application of friction and ocean pressure boundary conditions needs to be made, particularly by comparison with real-world geometries . Further work is needed to extend well-posedness analysis and error estimation in relation to the time-dependent grounding-line contact problem . The applicability of full Stokes ice-sheet simulations will need to be tested beyond the geometries and model set up presented in this study.