Time-dependent simulations of ice sheets require two equations to be solved: the mass transport equation, derived from the conservation of mass, and the stress balance equation, derived from the conservation of momentum. The mass transport equation controls the advection of ice from the interior of the ice sheet towards its periphery, thereby changing its geometry. Because it is based on an advection equation, a stabilization scheme needs to be employed when solved using the finite-element method. Several stabilization schemes exist in the finite-element method framework, but their respective accuracy and robustness have not yet been systematically assessed for glaciological applications. Here, we compare classical schemes used in the context of the finite-element method: (i) artificial diffusion, (ii) streamline upwinding, (iii) streamline upwind Petrov–Galerkin, (iv) discontinuous Galerkin, and (v) flux-corrected transport. We also look at the stress balance equation, which is responsible for computing the ice velocity that “advects” the ice downstream. To improve the velocity computation accuracy, the ice-sheet modeling community employs several sub-element parameterizations of physical processes at the grounding line, the point where the grounded ice starts to float onto the ocean. Here, we introduce a new sub-element parameterization for the driving stress, the force that drives the ice-sheet flow. We analyze the response of each stabilization scheme by running transient simulations forced by ice-shelf basal melt. The simulations are based on an idealized ice-sheet geometry for which there is no influence of bedrock topography. We also perform transient simulations of the Amundsen Sea Embayment, West Antarctica, where real bedrock and surface elevations are employed. In both idealized and real ice-sheet experiments, stabilization schemes based on artificial diffusion lead systematically to a bias towards more mass loss in comparison to the other schemes and therefore should be avoided or employed with a sufficiently high mesh resolution in the vicinity of the grounding line. We also run diagnostic simulations to assess the accuracy of the driving stress parameterization, which, in combination with an adequate parameterization for basal stress, provides improved numerical convergence in ice speed computations and more accurate results.

Numerical modeling is routinely used to understand the past and future
behavior of the ice sheets in response to the evolution of the
climate (e.g.,

It is well known that the discretization of advection-dominated equations by
the traditional finite-element method leads to numerical instabilities and
spurious oscillations

The finite-element method's literature presents a large number of
stabilization schemes, with different levels of complexity and
accuracy

The stress balance is another critical component of transient models. For
simplified stress balance equations, such as the shallow shelf
approximation

This is a consequence of the finite-element discretization, since such models employ continuous shape functions to approximate the ice thickness.

. This assumption implies that, for the grounded part of the element crossed by the grounding line, the ice surface is a function of both bedrock elevation and ice thickness, while, for the floating part, the ice surface is obtained by the hydrostatic floatation only, which only depends on the ice thickness. This makes the gradient of the ice surface and the resulting driving stress discontinuous within the elements containing the grounding line. While there exist comparison studies for basal friction and basal melt parameterizationsIn this context, the present paper aims to (i) assess the response of
different stabilization schemes in transient simulations subject to ice-shelf
basal melt and changes in basal friction, and (ii) develop and assess a
sub-element parameterization for the driving stress. The numerical experiments
are based on the Marine Ice Sheet Model Intercomparison Project for plan-view
models (MISMIP3d) setup

The evolution of the ice thickness is described by an advection equation with
source terms on the right-hand side:

The weak formulation of Eq. (

The weak form (Eq.

In general, stabilization schemes may be seen as a consistent way of adding
terms to Eq. (

Both schemes are interpreted as an upwind-equivalent scheme employed in the
finite-difference method

In the SUPG
scheme

The most common definition of the stabilization term

In ice flows dominated by internal deformation rather
than basal sliding, the velocity field may be described by the shallow ice
approximation. Within this approximation, the thickness equation turns into a
(non-linear) diffusion equation

An alternative to defining

In 2-D, it is common to bound the CFL by one-half over the entire domain and simulation period.

, the term within parentheses in Eq. (In ISSM, we implement

Strictly speaking, DG is not exactly a stabilization
scheme in the sense of adding upwinding terms to Eq. (

In DG, the weak formulation is written in an element-wise fashion, and the
advection operator is integrated by parts such that
Eq. (

As seen in Sect.

The flux-corrected transport (FCT) scheme operates in the resulting algebraic
system of the traditional Galerkin discretization (i.e.,
Eq.

The scheme modifies the discrete form of Eq. (

It is also named as consistent mass matrix, since it contains all terms from the FEM discretization.

,The first step consists of turning Eq. (

Compared to the original system (Eq.

In order to improve the accuracy of the solution while still preventing
spurious oscillations, the second step of the scheme consists in adding an
anti-diffusive term to the right-hand side of Eq. (

In Eq. (

The residual vector

Note that

In the linear FCT algorithm, the solution

In ISSM, the Crank–Nicolson scheme (i.e.,

Once the anti-diffusive flux is obtained by Eq. (

We employ a semi-implicit finite-difference time-stepping scheme to solve the
temporal evolution of the ice thickness. This scheme involves a backward Euler
method

For FCT, a Crank–Nicolson method is employed for

The position of the grounding line in non-full Stokes models is generally
tracked with a level set condition based on a floatation
criterion

The second condition in Eq. (

The driving stress parameterization is based on recovering the ice surface, and
consequently its gradient, on the element

For shallow shelf approximation

In this section, we describe the idealized geometry experiments used to
evaluate the stabilization schemes and the proposed driving stress
parameterization. For the latter, we employ different parameterization schemes
for basal friction. The list of all the schemes tested is summarized in
Table

List of the numerical schemes analyzed in this work.

The numerical experiments are based on the MISMIP3d
setup

We define the initial grounding line position to be close to its steady-state
position. According to boundary layer

The numerical experiments are divided in two sets of analyses: (i) diagnostic
analysis and (ii) prognostic analysis. The diagnostic analysis consists of
solving the stress balance equations under different sets of sub-element
parameterization schemes (driving stress and basal friction;
Table

The basal melt applied in the second set of experiments is defined as
follows

In all prognostic experiments, the grounding line is free to migrate, and its
position over the simulation time is updated according to a hydrostatic
floatation criterion, following Eq. (

Constants and parameters used along the numerical experiments.

Mesh resolution and associated number of elements

In order to quantify the performance of the stabilization schemes with real
ice-sheet geometries and numerical setups, we run transient simulations
(prognostic analyses) of the ASE, which includes the
fastest glaciers of WAIS. The glaciers in the ASE are subject to high
ocean-induced melt rates and are prone to the marine ice-sheet instability
(MISI), a positive feedback of grounding line retreat and increased ice
discharge sustained by a retrograde bedrock
slope

Our ASE domain includes Pine Island, Thwaites, and neighboring glaciers
(Haynes, Pope, Smith, and Kohler glaciers). We use BedMachine Antarctica
v1

For this setup, we perform only transient simulations with different
stabilization schemes. All simulations start from the same initial condition
and are forced by a constant surface mass balance obtained from the regional
climate model

To compare the ice speed from different sets of sub-element parameterizations,
we compute the speed from a reference model. The reference model is based on a
triangular structured conforming mesh with resolution of 50

Ice speeds obtained by the diagnostic analysis along a flow line
(

Convergence of the ice speed at the grounding line (

Error convergence of the ice speed for the diagnostic analysis in

Upstream of the grounding line, all sets of parameterizations “approach” the
reference speed (

We compare the transient results using the volume above floatation changes
(

Since the initial ice-sheet profile (Eq.

No external forcing experiment: ice surface and ice base at the end of
the experiment (

No external forcing experiment: evolution of volume above floatation change
(

No external forcing experiment: convergence of

In the setup where basal melt is applied only to fully floating elements
(i.e., no melt on partly floating elements), models using artificial diffusion
and streamline upwinding schemes produce almost 4 times the VAF losses
observed in the control experiment (Fig.

Basal melt experiment (no melt on partly floating elements):
evolution of

Basal melt experiment (no melt on partly floating elements):
convergence of

Basal melt experiment: ice surface and ice base at the end of the
experiment (

Basal melt experiment (melt on partly floating elements):
evolution of

Basal melt experiment (melt on partly floating elements):
convergence of

When some basal melt is also applied to partly floating elements, all models
generate VAF losses higher than those generated with the previous basal melt setup
(Fig.

Virtually all stabilization schemes produce the same

Basal friction perturbation experiment: evolution of

Basal friction perturbation experiment: convergence of

Basal friction perturbation experiment: convergence of grounding line
positions at the end of the experiment (

Basal friction perturbation experiment: convergence of grounding line
positions (relative errors) at the end of the experiment (

To evaluate the performance of the stabilization schemes in real ice-sheet
simulations (i.e., the ASE setup), we compare the VAF changes obtained with
transient simulations employing the five schemes considered in this work. For
the SUPG scheme, we chose the stability parameter as defined by
Eq. (

In the experiment forced by the first basal melt scenario (i.e.,

Under a higher basal melt scenario
(

Considering the entire ASE domain, in simulations forced by a low melt rate
(

The diagnostic analysis using the analytical ice-sheet profile
(Sect.

Steady-state GL positions for the MISMIP3d setup using
different sub-element parameterization schemes. GL positions for SEP1

Convergence of steady-state grounding line (GL) positions (

Employing an analytical expression of ice geometry based on a predefined
grounding line position allows the setup of reference models (i.e., models
whose mesh captures the exact position of the grounding line), in which no
errors due to parameterization schemes are introduced during the stress
balance solution (diagnostic analysis). Therefore, using the reference setup
improves the confidence of this analysis. Comparing grounding line positions
at steady state is another approach (Table

The prognostic analysis performed with the MISMIP3d-type geometry shows that
the numerical damping produced by the artificial diffusion and streamline
upwinding schemes impacts the accuracy of grounding line dynamics mainly in
simulations when large

For the prognostic analysis performed with real glaciers in West Antarctica
(Sect.

The choice of the stabilization scheme relies on the balance between
stability, accuracy, computational cost, and implementation effort. Yet, the
“best” choice could be never
reached

Apart from differences observed in terms of accuracy (i.e., VAF change), the
remaining differences between the stabilization schemes used here are their
numerical implementations and computational costs. The implementation of the
artificial diffusion, streamline upwinding, and SUPG is straightforward in
most of ice-sheet FEM-based models. However, the definition of the stability
coefficient for SUPG (Eq.

The convergence error of ice speed depends on the combination of
parameterizations chosen for basal friction and driving stress. Given that the
a priori error estimate is

We compute the weights

We note that

Six examples of structured and unstructured meshes are shown in Fig.

Examples of meshes employed in this work. Left panels are structured
conforming meshes, and right panels the unstructured meshes. Three mesh resolutions are
shown:

The numerical schemes evaluated here are currently
implemented in the ISSM. The code can be downloaded, compiled, and executed
following the instructions available on the ISSM website:

All data sets used in the prognostic analysis of the Amundsen Sea Embayment,
Sects.

The supplement related to this article is available online at:

MM and HS implemented some stabilization schemes in ISSM. TDS implemented the driving stress parameterization and stabilization SUPG. TDS designed the experimental setup and performed the simulations. TDS, MM, and HS led the analysis of the results. TDS led the initial writing of the paper. All authors contributed to writing the final version of the paper.

The authors declare that they have no conflict of interest.

This work is from the PROPHET project, a component of the International Thwaites Glacier Collaboration (ITGC). Support comes from the National Science Foundation. This is ITGC contribution no. ITGC-022. Hélène Seroussi is funded by grants from the NASA Cryospheric Sciences Program. We thank the editor, Steven Phipps, and the two reviewers, Stephen Cornford and Daniel Martin, for their positive and constructive comments, which improved the clarity of the manuscript.

This research has been supported by the National Science Foundation (grant no. 1739031).

This paper was edited by Steven Phipps and reviewed by Stephen Cornford and Daniel Martin.