Numerical glacier and ice-sheet models have been widely used in the context of climate change e.g.,, paleoglacial reconstruction e.g.,, and long-term landform evolution e.g.,. While models of ice dynamics typically include a multitude of processes beyond the flow of ice (e.g., meltwater flow and heat transport), the latter is typically by far the computationally most expensive part.

Three-dimensional simulations of the Stokes equations with a free surface (as implemented, e.g., in the model Elmer/Ice) and the shallow ice approximation (SIA) are end-members in the hierarchy of ice-flow models. In its simplest form, the SIA reduces the problem to a single partial differential equation for the elevation of the ice surface by assuming hydrostatic pressure and neglecting all stresses arising from horizontal shearing. Mathematically, the SIA requires that the ice layer is thin compared to the horizontal length scale defined by the topography. In particular for alpine glaciers, this condition is typically not met. In order to overcome this limitation, several depth-integrated approaches that account for horizontal stress components were developed, such as the second-order approximation iSOSIA proposed by and the Parallel Ice Sheet Model (PISM), which includes the shallow shelf approximation .

Several benchmark experiments were performed in order to compare depth-averaged models to solutions of the full Stokes equations based on synthetic examples e.g., or real-world scenarios e.g.,. These experiments confirmed the relevance of the theoretical limitations of the SIA concerning the topography and also pointed out its limited applicability to scenarios with fast sliding along the bed.

The numerical expense of all these models depends strongly on spatial resolution. achieved a spatial resolution of 1 km in a simulation covering the entire Alps (900×600 km) over a time span of 120 000 years with PISM. However, these simulations required high-performance computing facilities and a resolution of 1 km is by far not perfect for a complex topography with many valleys of various sizes. achieved a spatial resolution of about 500 m in 3-D full Stokes simulations of the Rhine glacier with Elmer/Ice. Although the domain was smaller and the simulated time span was much shorter then in the study of , these numbers tentatively suggest that the numerical advantage of depth-averaged models over full Stokes simulations is not as huge as it might be expected.

The limited computational performance of depth-integrated models arises from a combination of assuming hydrostatic pressure and the explicit time-stepping scheme implemented in almost all contemporary models . The main problem is that local variations in the slope of the ice surface generate a horizontal pressure gradient in the ice column down to the bed and thus affect the velocity of the entire ice column. As a consequence, all depth-integrated models, including the SIA in its simplest form, are susceptible to short-wavelength oscillations in surface elevation. Explicit schemes can only avoid the resulting instability by small time increments, which decrease quadratically with decreasing grid spacing.

The limited progress concerning the numerical efficiency has motivated the development of alternative approaches. In the context of large-scale landform evolution, and independently developed an approach to simulate erosion of valley glaciers without simulating the flow of ice explicitly. Implemented in the landform evolution model OpenLEM, it allows for simulations with spatial resolutions of less than 100 m over millions of years, and a benchmarking with the iSOSIA model yielded promising results . However, this approach circumvents the numerical issues of simulating the flow of ice for a specific application, but does not solve them.

The most remarkable alternative approach is based on deep learning. The Instructed Glacier Model (IGM) is an ice flow emulator that uses a Convolutional Neural Network trained by a two- or three-dimensional fluid-dynamical model. It achieves a speedup of up to three orders of magnitude compared to the original models. Recently, a version has been developed that even does not require a numerical model for training, but integrates a Convolutional Neural Network directly in the numerical solver .

The purpose of this paper is to challenge the hypothesis that the potential for further improvements in computational efficiency with classical numerical methods is limited . Despite the deficiencies of the SIA, this study only considers the SIA without extensions by horizontal stress components. The reason for this restriction is that the SIA can be formulated as a single partial differential equation, which makes the development of efficient numerical schemes much easier than for systems of differential equations. In this sense, the development proposed here is only a starting point with the perspective that parts of it can be transferred to models with a more elaborate stress balance in the future.

The approach presented in the following combines a semi-implicit time-stepping scheme as already implemented in a few models with a novel dynamic stabilization against small-wavelength oscillations. Implementations in MATLAB and Python are available. Since the MATLAB implementation was developed prior to the Python implementation, the numerical tests presented in this paper were performed with the MATLAB version. Further details are given in Sect. .

MinSIA simulates the flow of ice on a given topography b(x,y) in a two-dimensional approximation in Cartesian coordinates. The vertical ice thickness h(x,y,t) is the considered variable. The rate of change in ice thickness is defined by the volumetric balance equation 1∂h∂t=-divq+r where q(x,y,t) is the two-dimensional flux density (volume per time and cross-section length), r(x,y,t) the net rate of ice production and melting (thickness per time), and div the two-dimensional divergence operator.

Combining efficiency with maximum flexibility was the design goal of MinSIA. At the lowest level of complexity, simple predefined functions for q and r are offered, which do not require much additional programming for running the model. These functions can be combined with spatially and temporally variable parameter values for more complex applications. Finally, custom functions can be included, which also allow for coupling MinSIA with other models., e.g., for temperature.

The simple predefined model for the flux density is based on a decomposition into ice deformation and sliding along the bed, 2q=qd+qs. The contribution of deformation is described by the relation 3qd=-fdh5(cos⁡β)8|∇s|2∇s, where 4s(x,y,t)=b(x,y)+h(x,y,t) is the elevation of the ice surface and ∇s is its gradient. This relation is based on Glen's flow law, which assumes that the strain rate is proportional to τn with n=3, where τ is the shear stress. For a detailed discussion of this relation and the value of the exponent n, readers are referred to . The factor fd mainly depends on temperature in reality.

The factor involving the cosine of the slope angle β of the ice surface with 5cos⁡2β=11+tan⁡2β=11+|∇s|2 is a simple correction for steep slopes. It takes into account that the thickness perpendicular to the surface is lower than the vertical thickness and that driving forces by gravity are proportional to sin⁡β instead of tan⁡β=|∇s|. The correction is explained in the derivation of Eq. () in Appendix . It compensates the apparent inadequacy of the SIA for thin layers at steep bedrock slopes reported by Fig. 5. However, it is less elaborate than the full version of the SIA for an inclined bed developed by . While the full version allows for moderately different slopes of the bed and the ice surface, the simplified version considered here assumes that the bed is parallel to the ice surface. This simplified version is basically the same as the parallel sided slab elaborated by Sect. 7.2.

For sliding at the bed, the relation 6qs=-fsh3cos⁡β5|∇s|2∇s, is used. It combines the ideas of and see alsoChap. 7. As a strong simplification, Eq. () with a given value of the parameter fs relies on the assumption that the water pressure at the bed is proportional to the normal stress, following some simple models of glacial erosion that do not consider meltwater hydraulics explicitly . A short derivation of Eq. () is given in Appendix . It should, however, be emphasized that modeling sliding along the bed is still one of the major challenges in this field and that the applicability of the SIA is limited under fast sliding conditions e.g.,.

To make the model more flexible, the values of fd and fs may vary in space and time. This would, e.g., allow for taking into account their dependence on temperature in combination with a thermal model or taking into account water pressure explicitly.

The generic form of the flow law, 7q=-D(h,|∇s|)∇s, with a user-defined function D(h,|∇s|), offers the highest degree of flexibility.

Taking into account that ∂h∂t=∂s∂t, the considered differential equation then reads 8∂s∂t=divD(h,|∇s|)∇s+r, where the specific version defined by Eqs. () and () assumes 9D(h,|∇s|)=fdh5(cos⁡β)8+fsh3cos⁡β5|∇s|2. The property D is denoted diffusivity in the following, although the diffusion equation formally only allows a dependence of the diffusivity on s (and thus on h), but not on ∇s.

Different approaches are also included for the net rate of ice production and melting, r(x,y,t). The simplest model is piecewise linear with an equilibrium line altitude (ELA) ze and a maximum rate rmax⁡, 10r(z)=min⁡g+z-ze,rmax⁡forz≥zeg-z-zeforz<ze applied to the ice surface (z=s(x,y,t)), where g+ is the accumulation gradient (net rate of ice production per meter above the ELA) and g- is the ablation gradient (net rate of melting per meter below the ELA). A typical application would be a time-dependent ELA while keeping the other parameters constant. However, MinSIA allows for a variation of all parameter values in space and time as well as for replacing Eq. () by a custom function. Alternatively, r(x,y,t) can also be defined directly for each cell of the grid at each time.

The present-day topography of the Southern Black Forest was used for the numerical tests. The considered domain consists of a 60×60 km square centered to the highest peak (Feldberg, 1493 m). The publicly available 1 m terrain model was downsampled to a grid spacing of δx=δy=25 m, resulting in a grid of 2400×2400 cells. The spatial resolution was chosen to investigate the accuracy and the efficiency of the numerical scheme within in the framework of the SIA. To the author's knowledge, such a high spatial resolution cannot be achieved at a reasonable computational effort by any existing scheme for the SIA or for a more elaborate model of ice flow. In turn, the considered topography contains several steep valleys where the physical limitations of the SIA become relevant. This topic will be briefly addressed in Sect. , but is not the primary subject of this paper.

The values fd=5.34×10-5 m-3yr-1 and fs=3.56 m-1yr-1 are assumed for the flow parameters, corresponding to the values used by . For the net rate of ice production, the simple ELA-based approach (Eq. ) was used with an accumulation gradient of g+=0.002 yr-1 and an ablation gradient of g-=0.003 yr-1. These values are oriented on the rates assumed in the model comparison of . All simulations were run over a time span of 3000 years with a constant ELA at ze=1000 m, starting from an ice-free surface. This setup does not address anything that ever happened, but was designed as a simple scenario with a transition from isolated valley glaciers towards a continuous ice cap, as illustrated in Fig. .

Time increment was δt=1/64 years and the smoothing factor was set to f=0.25 as a reference. As the maximum thickness achieved at the end was h=637 m, this choice corresponds to smoothing over squares of 13×13 cells (325 m; m=6 in Eq. ) at maximum. While it would be desirable to use a simulation without smoothing as a reference scenario, the scenario considered here was originally chosen as a tradeoff between accuracy and numerical effort. The semi-implicit scheme without smoothing requires δt=1/4096 years and would have taken several months on a standard desktop PC. The error arising from the choice of the reference scenario will be discussed at the end of Sect. .

All simulations, except for the tests of the linear solver (Sect. ), were performed either with the direct solver or with the PCG solver with a relative tolerance of 10-10. This value is small enough to ensure that the errors caused by the iterative PCG solver are negligible.

Finding a suitable time increment is the main challenge in setting up a simulation with MinSIA. As illustrated for the considered example in Fig. , the error caused by the finite time increment is quite small as long as staircase oscillations are avoided. So a time increment slightly below the limit at which staircase oscillations are initiated is typically the best tradeoff between accuracy and efficiency.

The findings from Sect.  suggest that the dependence of the maximum time increment on grid spacing is weaker than linear. Since the criteria that are typically applied to explicit schemes predict a stronger dependence, these criteria are not immediately useful for estimating the maximum time increment. In particular, the Fourier criterion for the diffusion equation predicts a maximum δt proportional to δx2, which would be a much stronger dependence than found here numerically. The Courant–Friedrichs–Lewy (CFL) criterion for the advection equation, which roughly says that the material must not move by more than the grid spacing in one step, predicts a maximum δt proportional to δx, which also is still stronger than found here numerically. The maximum depth-averaged velocity is approximately 500 myr-1 here, which means that distances of up to 5δx are traveled per step for δx=25 m and δt=0.25 years. So the CFL criterion is also exceeded absolutely.

However, the CFL criterion becomes relevant in some situations. As described in Sect. , The numerical scheme was tailored to capture the high sensitivity of D(h,|∇s|) to changes in |∇s| at large ice thickness h. In turn, the sensitivity to changes in h is high at steep slopes. Concerning this sensitivity, the semi-implicit scheme is not much better than an explicit scheme.

Figure  illustrates the weakness of the scheme at steep slopes for δx=25 m and δt=0.25 years. Oscillations of several meters occur during the early phase when the maximum thickness is still much lower than at the end. A detailed analysis of the strong oscillations with δh≥10 m around t≈50 years reveals that these affect only a few nodes. While the slope |∇s| is greater than 0.74 at these nodes, the ice thickness is lower than 30 m.

These oscillations are not a problem for the test scenario. Practically, it does not matter whether ice flows down steep slopes uniformly or in distinct surges, provided that the oscillations do not cause large errors in the volumetric balance due to negative values of h. In this example, the oscillations even cease for t>100 years due to filling up the steep slopes with ice. However, this is not necessarily the case for other topographies.

Some additional, but preliminary tests were performed for the Rhone–Aare region in Switzerland and the Salzach region in Austria as two examples of alpine topography. All parameter values were the same as for the Black Forest example, except for an ELA of ze=1500 m and an upper limit for the net rate of ice production (Eq. ) of rmax⁡=1 myr-1.

A time increment of δt=1/32 years turned out to be sufficient for both scenarios at δx=50 m, whereby the maximum ice thickness exceeded 2000 m in the Rhone–Aare example and 1450 m in the Salzach example. This time increment is 8 times smaller than that found for the Black Forest example. However, the maximum ice thickness is also more than 3 times higher and it was already found in Sect.  that the occurrence of staircase oscillations depends on ice thickness.

However, larger time increments caused oscillations of several meters at steep slopes already in an early phase of the simulation at moderate ice thickness. These oscillations cease in the Salzach example when the valleys have been filled with ice, which allows for increasing δt to 1/16 years. This is, however, not the case for the Rhone–Aare example.

So the numerical efficiency is limited by the weak performance at steep slopes for moderate ice thickness in the alpine examples. Even increasing the ELA so that the maximum ice thickness is similar to that in the Black Forest example does not bring the maximum time increment close to the value δt=1/4 years achieved there.

Since the scheme is not much better than an explicit scheme concerning the dependence of the ice flux on h at steep slopes, the CFL criterion becomes relevant here in contrast to the results from the previous section. As mentioned above, the CFL criterion predicts a stronger (linear) dependence of the maximum δt on δx than found numerically for large ice thickness. Therefore, the limitation arising from steep slopes becomes more relevant with increasing spatial resolution.

As a consequence, simulations of large alpine regions with δx<50 m are hardly feasible on standard PCs. Beyond the limitation of the maximum time increment, the number of iterations of the PCG scheme also contributes to the increasing numerical effort. The results on the tolerance of the PCG solver found in Sect.  also hold for the alpine examples, whereby a tolerance of 10-5 even seems to be slightly better than in the Black Forest example. However, the absolute number of iterations is higher for the alpine examples. It even reaches 100 at h=2000 m, which lets the PCG solver contribute more than 80 % to the total effort.

However, the results obtained from the numerical tests still do not allow for estimating the maximum time increment from the topography and the parameters directly. Finding the maximum time increment experimentally by watching the maximum oscillation δh (Eq. ) seems to be the best way to estimate the maximum δt at present.

While the numerical scheme proposed here introduces a major progress concerning the numerical efficiency, it cannot overcome the physical limitations of the SIA. As a main limitation, the SIA in the simple form considered here completely neglects stresses arising from horizontal shearing.

As an example, Fig.  shows the valley cross section marked in Fig. d. The valley is quite narrow here with a large ice thickness. The slope angles are about 30° and the depth-to-width ratio is higher than 0.25 and thus clearly not close to zero as theoretically assumed for the SIA.

The simulated flow velocity is strongly concentrated around the center of the valley. The depth-averaged velocity decreases to half of its maximum value over a distance of about 150 m. In relation to a maximum thickness of about 300 m, this means that the horizontal velocity gradients are in the same order of magnitude as the vertical gradients. Accordingly, the horizontal shear stresses not captured by the SIA are in the same order of magnitude as the vertical shear stress and are by far not negligible.

In combination with the large ice thickness, the horizontal shear stresses would slow down the flow strongly in the central region. In turn, the horizontal stresses exert a driving force to regions of low velocity and thus increase the velocity there. Furthermore, these stresses increase the fluidity of the ice according to the nonlinear flow law. So the SIA overestimates the velocities in the central part of the valley and vice versa.

As shown in Fig. , neither the ice surface nor the velocities depend strongly on δx at least for δx≤100 m. The variations in the elevation of the ice surface at the center are a few meters, in agreement with the overall deviations analyzed in Fig. . In combination with the strong effect of horizontal stresses, the moderate dependence of thickness and velocity on δx shows, however, that we are already in the regime where the physical error of the SIA dominates over the numerical error.

The artificial concentration of high velocities around the center of the valley would be a crucial issue in glacial erosion models that assume that the erosion rate depends on the local sliding velocity e.g.,. Here it would even generate narrow canyons instead of U-shaped valleys, and their width would depend on the spatial resolution. In turn, the effect on reconstructions of ice margins might be smaller since the errors in flow velocity compensate partly and small changes in ice thickness have a large effect on the ice flux. So it depends strongly on the application how far the SIA can be pushed outside its theoretical range of validity, defined by the ice thickness being much smaller than the length scale of horizontal variations.

While the physical deficiencies of the SIA obviously limit the merit of spatial resolutions much finer than the width of steep valleys, this is not necessarily the case for the entire topography. Figure  shows the profile through the highest peak marked in Fig. d. The ice thickness is much smaller than the horizontal scale of the major topographic structures in the central part (e.g., 18 m at the highest peak and 7.7 m at the minimum). Since either the thickness is low or the spatial variation in velocity is moderate in each part of the profile, horizontal stresses should not have a major effect here. So the limitations of the SIA should not be crucial here.

In any case, it would be desirable to include horizontal stress components. It is, however, not trivial whether any of the existing approaches (a second-order approximation or a combination with the shallow shelf approximation) could be implemented in such a way that the numerical efficiency is preserved or whether a different approximation is needed.

At present, MATLAB and Python implementations of MinSIA are available under the GNU General Public License. None of them requires specific packages, except for NumPy and SciPy for the Python version.

For performance reasons, the PCG solver and the incomplete Cholesky preconditioner used in the Python version were written in C++. Depending on the operating system, the C++ code (pcg.cpp) has to be compiled and the respective shared object file (pcg.so) should be placed in the same folder as the Python package (minsia.py). The respective dynamic-link library for Windows systems (pcg.dll) can be directly downloaded instead of compiling the source code.

Developers are encouraged to include and test alternative solvers for the linear equation system, which typically consumes the biggest part of the computing time. In particular for the Python version, it is recommended to include alternative solvers at the location of the direct solver in the code since the sparse matrix and the right-hand side of the equation system are easily recognized there. Improving the performance further by GPUs may be one option. It should, however, be noted that the standard PCG solver of SciPy was by up to 20 times slower than the compiled C++ version in the tests, which may reduce the improvement achievable by parallel computing strongly.

The implementation is minimalistic and consists of a class minsia (MATLAB) or MinSIA (Python), which contains only a constructor and a method step for performing a forward time step. Data handling and visualization are left to the user.

The constructor requires 4 mandatory arguments:

An array b for the topography b.

This paper presents a novel numerical scheme and a lightweight implementation of the SIA for the flow of ice over a given topography. Key features are a semi-implicit time-stepping scheme and a dynamic smoothing of the slope of the ice surface. The proposed scheme overcomes the limitations of explicit schemes at large ice thickness and small slopes, which are crucial to their numerical efficiency. The results of the performed numerical tests with grid spacings down to 25 m are promising.

Numerically, steep slopes are the main problem. In contrast to thick ice layers with moderately inclined surfaces, the new scheme is not much better than an explicit scheme for thin layers on steep slopes. This limitation depends on the topography and on the model parameters and becomes increasingly relevant for finer grids. Practically, some loss of performance was observed for steep alpine topographies at 50 m grid spacing. However, this loss does not affect the improvement compared to explicit schemes strongly.

Conceptually, the restriction to the SIA is the main limitation of the new scheme at present. Since the SIA neglects all horizontal stress components, it overestimates the flow velocities in the middle of valleys and vice versa. The SIA is also not well-suited if sliding along the bed is strong.

The physical limitation due to the restriction to the SIA is clearly stronger than the remaining numerical limitations. The new scheme allows for spatial resolutions that seem not to be achievable with conventional numerical models at a reasonable computational effort. In turn, however, such high resolutions are particularly useful in rugged alpine terrain, where the limitations of the SIA are most severe. At the moment, it has to be evaluated carefully how much the physical deficiencies of the SIA affect the respective application.

In this sense, challenging the hypothesis about the limited potential for further improvements in computational efficiency with classical numerical methods made by was partly successful. Numerical performance has been improved greatly, but only for a very simple model with clear physical limitations. Therefore, the approach developed here cannot compete with the Instructed Glacier Model (IGM) at present since the IGM achieves a high numerical efficiency without obvious limitations concerning the underlying physics.

Concerning the future development, the crucial question is whether horizontal stress components can be included to overcome the limitations of the SIA. If this is possible without losing much of the numerical efficiency, classical numerics may become competitive again to the IGM. It is, however, not trivial whether any of the existing approaches (a second-order approximation or a combination with the shallow shelf approximation) could be implemented in such a way that the numerical efficiency is preserved or whether a different approximation is needed.