Viscous, gravity-driven, free surface problems are prevalent in geoscientific applications such as mantle convection and glaciology . Such simulations involve solving the Stokes equations for the velocity and pressure, (u(x,y,z,t), p(x,y,z,t)), and a free-surface equation for the upper surface height h(x,y,t) that determines the geometry, see Fig. . A key computational challenge arises from the strong coupling between the evolving geometry and the velocity field, which can lead to numerical stability issues and imposes severe restrictions on the time-step size, ultimately reducing computational efficiency. Fully implicit methods would overcome these time-step limitations, but the strong coupling introduces convergence issues that prevent practical usage.

In ice-sheet and glacier modeling, the time-step size restriction is a prominent problem . Ice-sheet and glacier models describe the movement of the ice covering Antarctica and Greenland, or that of smaller mountain glaciers and ice caps. Under the long time scales that are relevant to ice dynamics, ice deforms as a viscous fluid. It slowly creeps under its own weight from the interior of a continent to the coast, where it melts due to lower altitudes and meeting the warm ocean. Accurately calculating the geometry and the rate of flow towards the ocean is crucial in estimating the future rise in sea level in response to climate change. However, the computational challenges of ice-sheet modeling limit the accuracy of projections . Due to the coupling between geometry and ice velocity, the full problem can be likened to solving the heat equation, but the state-of-the-art numerical methods are explicit in time. To maintain stability, time-steps are therefore often limited to around 0.1–1.0 years , which is a bottleneck. Implicit iterations would increase the largest stable time-steps but are challenging and have so far only been achieved for approximate models . For models based on the full Stokes equations there is, to our knowledge, no working implicit method based on iterations. A monolithic implicit solver was constructed in for geodynamics. The great promise of implicit methods was recognized by who later conjectured that such schemes should be well-posed .

In this paper, we present a numerically stable and accurate fully implicit method by introducing boundary terms in the Stokes equations, which vanish during convergence. We argue that the convergence issues of standard implicit iterations are connected to the stability issues that are seen for explicit methods (since the first iteration has to be explicit), and we introduce a method that circumvents these issues in a computationally efficient way. The method relies on the Free Surface Stabilization Algorithm (FSSA), which was originally developed for mantle convection problems and later adapted for ice sheet simulations . The FSSA relaxes the time-step restrictions for explicit solvers by adding additional boundary terms. We adapt this method to the context of fully-implicit solvers. The boundary terms vanish as convergence is reached. This method allows for very large time-steps and ensures consistency. We then utilize the new implicit method to construct second-order accurate time-stepping schemes and demonstrate efficiency and accuracy on setups relevant to geodynamics and glacier simulations.

The surface height position h, the velocity u=(ux,uy,uz) and the pressure p are the solution to a system of coupled partial differential equations, namely the p-Stokes-coupled free-surface equation: 1-∇⋅S(Du)+∇p=ρg,∇⋅u=0inΩ(t),2∂h∂t+u(x,y,z=h,t)⋅∇h=asonω. Here, Ω⊂Rd with d∈{2,3}, is an open and bounded domain with a boundary Γ, of which one part is the upper free surface Γs which is located at a height h, see Fig.  for an illustration from glaciology. The free-surface equation (Eq. ) is solved on ω∈Rd-1, which represents a projection of the upper ice surface Γs onto a flat two-dimensional plane. The ice-flow dynamics are driven by the gravitational term ρg. The free-surface equation can be forced by the term as, which in a glaciological context represents accumulation and ablation. The deviatoric stress tensor S depends on the strain-rate tensor 2Du=∇u+(∇u)T according to a constitutive law 3S(u)=ν(Du)Du, where 4ν:=ν(Du)=ν0ε2+|Du|2p-22p∈(1,2], is the viscosity function, dependent on the effective strain rate |Du|; ε2 is a regularisation term to avoid infinite viscosity at zero shear strain rate. The relation is linear for p=2, as for Newtonian dynamics and non-linear for non-Newtonian materials such as glacial ice for which p=4/3, where p is usually stated in terms of the Glen's parameter n=1/(p-1). We consider a very viscous flow, so that ν0 is large.

A key challenge in the time discretization of such problems arises from the fact that the computational domain Ω is coupled to the time-dependent height of the ice surface h(x,y,t) so that Ω(t)=Ω(h(t)). The dependence of the domain on the surface elevation h means that the velocity field also implicitly depends on h. At the same time, these velocity components appear as coefficients in the free surface evolution equation that governs the changes in h. The mutual dependence between h and u limits the numerical stability of explicit methods.

The Stokes equations are solved under the following boundary conditions: At the top surface Γs a stress-free boundary condition is applied 5(S(u)-Ip)⋅n=0onΓs. Here, n=(nx,ny,nz) denotes the outward pointing normal and I the identity matrix. The lateral boundaries, ΓW and ΓE (see Fig. ), are assumed to be rigid in this study so that an impenetrability condition applies, 6u⋅n=0,onΓW∪ΓE. Lastly, at the base or bedrock, Γb, we apply either a no-slip condition 7u=0onΓb, or an impenetrability condition combined with a so-called Weertman sliding law 8ti⋅(σn)=-Cu⋅tionΓb, where ti are tangent vectors that span the lower ice surface and C is the friction coefficient. Note that the methods in this article are not restricted to the specific boundary conditions on Γb, ΓW and ΓE.

We consider a finite element discretization in space, which is commonly used in both ice sheet modeling and geodynamics . The weak formulation of the Stokes problem is to find the velocity and pressure (u,p)∈[W01,p(Ω)]d×Lp′(Ω) so that 9(S(u),Dv)Ω-(∇⋅v,p)Ω-(∇⋅u,q)Ω=(ρg,v)Ω∀(v,q)∈W01,p(Ω)d×Lp′(Ω), where (⋅,⋅)Ω denotes the L2 scalar product, e.g. (u,v)Ω=∫Ωu⋅vdx. The appropriate spaces for the continuous problem are the Sobolev space [W01,p(Ω)]d and the Lebesgue space Lp′(Ω), where p′=p/(p-1) e.g.,.

In this work, we discretize the weak form using Taylor-Hood elements, i.e. linear elements for the pressure and quadratic elements for the velocity. Such a discretization is inf-sup stable .

The weak form of the free surface equation is to find the surface height h∈W1,2(ω) such that 10∂h∂t,qω=-ux∂h∂x-uy∂h∂y+uz,qω,∀q∈W1,2(ω). In this paper, it is useful to note that the equation can be written equivalently as 11∂h∂t,qω=∫Γsu⋅nqds. The equivalence is easily seen by changing from integration on the surface to the two-dimensional footprint ω, and considering that the normal vector pointing out of a surface is n=ñ/‖n‖, where ñ=(-∂xh,-∂yh,1). This derivation takes the form 12(u⋅n,q)Γs=∫Γsu⋅nqds=∫ωu⋅nq1+∂xh2+∂yh2dx=∫ωu⋅-∂xh,-∂yh,111+∂xh2+∂yh2q1+∂xh2+∂yh2dx=∫ωu⋅-∂xh,-∂yh,1qdx=∫ω∂h∂tqdx. The free surface equation is discretized using linear elements.

The standard approach to solving the coupled system (Eqs.  and ) is to solve the Stokes equations once per time step on a mesh that reflects the geometry from the preceding step, (Algorithm 1). An explicit Euler discretization of the free surface can thus be written as 13hk+1,qω=hk,qω+Δt-uxk∂hk∂x-uxk∂hk∂x+uzk,qω=hk,qω+Δt∫Γskuk⋅nkqdsk,∀q∈W1,2(ω), where k is the current time step and k+1 is the next time step. In this explicit Euler discretization, the velocity, u, is, by necessity, always evaluated on time step k. It is easy to change to an implicit treatment of the surface height, h, by evaluating it at step k+1 instead, but as was shown in , it is the treatment of the velocity u that determines the stability properties of the overall system. The explicit treatment of velocities is essentially analogous to solving the heat equation using an explicit Euler time-stepping scheme, and therefore, the time-step size is limited by stability constraints rather than by accuracy considerations.

To avoid the restriction on the time-step, an implicit solver could in theory be used. A natural approach to such a solver is to iterate between the free surface solver and the Stokes solver until convergence. Unfortuntaely, the iterations do not converge for large time-steps. Here we adress this issue.

To relax the stability restriction without implicit iterations, the FSSA method was introduced in geodynamics in and subsequently adapted to ice-sheet modeling in . One way of understanding the FSSA stabilization method is by applying the Reynolds transport theorem to the gravitational force on weak form, assuming that ∂∂t(ρg⋅v)=0, which leads to: 14∂∂t∫Ω(t)ρg⋅vdx=∫∂Ω(t)ρ(u(t)⋅n(t))(g⋅v)ds. An explicit Euler discretization of the above relation gives an expression for how the driving stress on time-step k+1 relates to the driving stress on time-step k: 15∫∂Ωtk+1ρg⋅vdx≈∫Ωtkρg⋅vdx+Δt∫∂Ωtkρ(u⋅n)(g⋅v)ds︸stabilization, where the last term is the FSSA stabilization term. Note that the original FSSA method of includes a parameter θ in front of the stabilization term which will not be used here. The stabilization is added to the Stokes equations (Eq. ) rather than the free-surface equation (Eq. ). Due to the impenetrability condition, the FSSA term vanishes on all surfaces except Γs. Since the flow is gravity driven, it appears to be sufficient to represent the force of gravity implicitly (which FSSA does approximately) to yield an implicit solution. Note, however, that it is not a true fully implicit solver. The FSSA method has proved remarkably successful in increasing the largest stable time step in marine ice sheet models, by up to a factor of approximately 100 .

An observation that will be useful for creating the fully implicit solver is that the FSSA term is similar to the right-hand side of the discretized free surface equation (Eq. ), the difference lies in the test function (which determines which equation the term is added to) and the scaling ρg which is only in the FSSA term.

A true fully implicit solver requires iterations between the free surface equation and Stokes equations in each time step, as in Algorithm .

Note that in step 6 of Algorithm 2 the time-discretized free surface equation is written on strong form for brevity of presentation, while in practice it is solved on weak form. Since for the first implicit iteration r=0, the Stokes equations is solved on Ω0k+1=Ωk, the first step is indeed explicit, i.e. (u0k+1,p0k+1)=(uk,pk).

In theory, a fully implicit solver should be unconditionally stable. However, in practice the implicit iterations (index r in Algorithm 2) do not converge for large time steps, making the fully implicit solver no more useful than an explicit one. This is perhaps not surprising, as the first iteration in the implicit loop always consists of an explicit step in the velocity-pressure solution, which can induce instability for larger time steps and renders a poor first guess for the velocity coefficients in the free-surface equation (see the line labeled Explicit Euler in Fig. b.

The main goal of this paper is to stabilize the implicit iterations using two slightly different FSSA terms subtracted from each other. At the first non-linear iteration, r=0, of each time step, an explicit FSSA-stabilized method is used just as for the purely explicit solver of Sect. . This renders a first guess that is close enough to the true implicit solution for the iterations to converge, this is the line labeled FSSA in Fig. b. For large time steps, the subsequent iterations must also be stabilized to avoid the growth of spurious oscillations, but applying the FSSA again would be erroneous since FSSA predicts the surface position at the next time step, not the next iteration. To create an appropriate stabilization term, we observe that the surface at implicit iteration r+1, i.e. hr+1k+1, is computed as 16hr+1k+1,qω=hk,qω+Δt-ux,rk+1∂hrk+1∂x-uy,rk+1∂hrk+1∂y+uz,rk+1,qω=hk,qω+Δt∫Γs,rkurk+1⋅nrk+1qdsrk+1, while the surface at the previous implicit iteration, hrk+1, is computed as 17hrk+1,qω=hk,qω+Δt-ux,r-1k+1∂hr-1k+1∂x-uy,r-1k+1∂hr-1k+1∂y+uz,r-1k+1,qω=hk,qω+Δt∫Γs,r-1kur-1k+1⋅nr-1k+1qdsr-1k+1. Therefore, the difference between two subsequent surfaces in the implicit iterations is 18hr+1k+1,q-hrk+1,q=Δt∫Γrkurk+1⋅nrk+1qdsrk+1-∫Γr-1kur-1k+1⋅nr-1k+1qdsr-1k+1, see Fig.  for a conceptual illustration. Recalling the similarity between the terms of the right-hand side of Eq. () and the expression above, we claim that the FSSA term must be augmented to be consistent with Eq. (). The FSSA term of the previous iterations is thus subtracted from the FSSA term on the current iteration, leading to: 19θ1Δt∫Γrkρg⋅vurk+1⋅nrk+1dsrk+1-θ2Δt∫Γr-1kρg⋅vur-1k+1⋅nr-1k+1dsr-1k+1. Here, we have introduced the parameters θ1 and θ2 for later convenience. We will call this method the subtraction-FSSA, and we will use Eq. () to stabilize the fully implicit discretization for r>0. By the subtraction-FSSA, the term Δtu⋅n in Eq. (), which is the net distance traveled by the surface from one time step to another, is replaced by Eq. () to ensure that the correct surface displacement is used in the Reynolds transport theorem. Setting θ1=1, θ2=0 results in the original FSSA term and, for the new subtraction-FSSA, θ1=θ2=1. Exactly like for the original FSSA, the subtraction-FSSA of Eq. () is added to the Stokes equations. A key observation is that, if the iteration converges 20hr+1k+1,q-hrk+1,q→0asr→∞, i.e. as the implicit discretizations converge, the difference between the surfaces at consecutive iterations tend to zero and thereby the stabilization also approaches zero. The iterations should therefore converge to the unstabilized implicit solution.

In some software programs, like Elmer/Ice, it is difficult to access the mesh and normal vector from iteration k, required in Eq. (). For this purpose we introduce a simplified subtraction-FSSA, which only requires the velocity to be accessible from the previous iteration 21θ1Δt∫Γrkρg⋅vurk+1⋅nrk+1dsrk+1-θ2Δt∫Γrkρg⋅vur-1k+1⋅nrk+1dsrk+1.

For first order Euler discretizations of the free surface equation, the numerical error increases linearly with the time step. The increased stability of FSSA and fully implicit solutions allows for large time steps, which can introduce large errors. To reduce errors, we now construct higher order time-stepping schemes. It is of course trivial to exchange the discretization of the free surface equation with a higher-order scheme. However, the FSSA stabilization term was derived from a first-order discretization of the Reynold's transport theorem, and higher-order effects of the h-dependency of terms other than the body force term were ignored. The FSSA-stabilization will thus reduce the overall accuracy to first order. However, with the new implicit scheme introduced in Sect.  it is possible to achieve higher-order accuracy since the stabilization term of Eq. () tends to zero. Thus, a standard implicit higher-order scheme can be applied to the free surface. Here, we choose a second-order Crank–Nicolson scheme as well as a second-order Backward Differentiation (BDF2) scheme. We present these schemes on strong form and drop the index r for brevity of presentation. The Crank–Nicolson scheme is 22hk+1=hk+Δt12-uxk∂hk∂x-uyk∂hk∂y+uzk+12-uxk+1∂hk+1∂x-uyk+1∂hk+1∂y+uzk+1 where uk+1 is the solution retrieved from the implicit iterations and uk is the solution of the previous time step. The BDF2-scheme is 233hk+1-4hk+hk-12Δt=-uxk+1∂hk+1∂x-uyk+1∂hk+1∂y+uzk+1.

We have developed a fully implicit method that allows for very large time-steps as well as second-order accuracy in simulations of viscous free-surface flow simulations. The method has important applications in e.g. ice sheet and glacier modelling and geodynamics. The implicitness allows for large time-steps and thereby simulation speedup, while second order schemes keep accuracy high. The key is a new stabilization scheme, the subtraction-FSSA, which extends the original FSSA approach of , and . Compared to the original FSSA stabilization, our new scheme increases accuracy, not only because it enables second-order time-stepping schemes but also because it allows for a more accurate representation of the free-surface constraint preventing negative thickness at ice sheet and glacier margins. We further find that a simplified, easy-to-implement variant of the implicit iterations performs equally well as the full method and requires only access to the velocity of the previous iteration.

For first-order schemes, our implicit solver matches the accuracy of explicit FSSA in unconstrained geodynamic problems, confirming that FSSA already approximates an implicit solution well in such cases. In contrast, for moving glacier fronts, implicit iterations are substantially more accurate. It is overall more computationally efficient to use large time steps with three implicit iterations compared to using smaller time-steps with no implicit iterations.

Second-order schemes improve accuracy for both unconstrained and constrained (glacier) problems, requiring only two implicit iterations for the former and three for the latter, keeping computational costs low. The approach of using implicit iterations to construct higher order schemes appears especially appealing for glacier and ice sheet simulations, where the implicit iterations are needed regardless for resolving the minimum thickness constraint with large time-steps. Implementations such as BDF2 add minimal overhead, needing only a small modification to the left-hand side of the free-surface equation (Eq. ). Overall, second-order implicit time-stepping with large time-steps is generally more efficient than first-order explicit schemes.

Future work will refine stopping criteria for implicit iterations and explore how to further increase accuracy near the glacier front, where a minimum thickness constraint is needed. None-the-less, subtraction-FSSA represents a major advance in accuracy and efficiency. Its integration into Elmer/Ice makes it readily available to the ice sheet community, paving the way for faster, more precise glacier and ice sheet simulations.