We capture the flow of an incompressible, non-linear, viscous fluid – including a temperature-dependent rheology. Since ice is approximately incompressible, the equation for conservation of mass reduces to 1∂vi∂xi=0, where vi is the velocity component in the spatial direction xi.

Neglecting inertial forces, ice's flow is driven by gravity and is resisted by internal deformation and basal stress: 2∂τij∂xj-∂P∂xi+Fi=0, where Fi=ρgsin⁡(α)[1,0,-cot⁡(α)] is the external force. Ice density is denoted by ρ, g is the gravitational acceleration and α is the characteristic bed slope. P is the isotropic pressure and τij is the deviatoric stress tensor. The deviatoric stress tensor τij is obtained by decomposing the Cauchy stress tensor σij in terms of deviatoric stress τij and isotropic pressure P.

In the absence of phase transitions, the temporal evolution of temperature in deforming, incompressible ice is governed by advection, diffusion and shear heating: 3ρc∂T∂t+vi∂T∂xi=∂∂xik∂T∂xi+τijϵ˙ij, where T represents the temperature deviation from the initial temperature T0, c is the specific heat capacity, k is the spatially varying thermal conductivity and ϵ˙ij is the strain rate tensor. The term τijϵ˙ij represents the shear heating, a source term that emerges from the mechanical model.

Shear heating could locally raise the temperature in the ice to the pressure melting point. Once ice has reached the melting point, any additional heating is converted to latent heat, which prevents further temperature increase. Thus, we impose a temperature cap at the pressure melting point, following , by describing the melt production using a Heaviside function χ(T-Tm): 4ρc∂T∂t+vi∂T∂xi=∂∂xik∂T∂xi+[1-χ(T-Tm)]τijϵ˙ij, where Tm stands for the ice melting temperature. We balance the heat produced by shear heating with a sink term in regions where the melting temperature is reached. The volume of produced meltwater can be calculated in a similar way as proposed by .

We approximate the rheology of ice through Glen's flow law : 5ϵ˙ij=12∂vi∂xj+∂vj∂xi=a0τIIn-1exp⁡-QR(T+T0)τij, where a0 is the pre-exponential factor, R is the universal gas constant, Q is the activation energy, n is the stress exponent and τII is the second invariant of the stress tensor defined by τII=1/2τijτij. Glen's flow law posits an exponent of n=3.

At the ice-top surface Γt(t), we impose the upper surface boundary condition σijnj=-Patmnj, where nj denotes the normal unit vector at the ice surface boundary and Patm the atmospheric pressure. Because atmospheric pressure is negligible relative to pressure within the ice column, we can also use a standard stress-free simplification of the upper surface boundary condition σijnj=0. On the bottom ice–bedrock interface, we can impose two different boundary conditions. For the parts of the ice–bedrock interface Γ0(t) where the ice is frozen to the ground, we impose a zero velocity vi=0 and thus no sliding boundary condition. On the parts of the ice–bedrock interface Γs(t) where the ice is at the melting point, we impose a Rayleigh friction boundary condition – the so-called linear sliding law – given by 6vini=0,niσijtj=-β2vjtj, where the parameter β2 denotes a given sliding coefficient, ni denotes the normal unit vector at the ice–bedrock interface and tj denotes any unit vector tangential to the bottom surface. On the side or lateral boundaries, we impose either Dirichlet boundary conditions if the velocities are known or periodic boundary conditions, mimicking an infinitely extended domain.

For numerical purposes and for ease of generalisation, it is often preferable to use non-dimensional variables. This allows one to limit truncation errors (especially relevant for single-precision calculations) and to scale the results to various different initial configurations. Here, we use two different scale sets, depending on whether we solve the purely mechanical part of the model or the thermomechanically coupled system of equations.

In the case of an isothermal model, we use ice thickness, H, and gravitational driving stress to non-dimensionalise the governing equations: 7L‾=H,τ‾=ρgL‾sin⁡(α),,v‾=2nA0L‾τ‾n, where A0 is the isothermal deformation rate factor and α is the mean bed slope. We can then rewrite the governing equations in their non-dimensional form as follows: 8∂vi′∂xi′=0,∂τij′∂xj′-∂P′∂xi′+Fi′=0,ϵ′˙ij=12∂vi′∂xj′+∂vj′∂xi′=2-nτII′n-1τij′, where Fi′ is now defined as Fi′=1,0,-cot⁡(α). The model parameters are the mean bed slope α and domain size in each horizontal direction, i.e. Lx′ and Ly′.

Reducing the thermomechanically coupled equations to a non-dimensional form requires not only length and stress, but also temperature and time. We choose the characteristic scales such that the coefficients in front of the diffusion and shear heating terms in the temperature evolution Eq. () reduce to 1: 9T‾=nRT02Q,τ‾=ρcpT‾,t‾=2-na0-1τ‾-nexp⁡QRT0,L‾=kρcpt‾. These choices entail the velocity scale in the thermomechanical model being v‾=L‾/t‾. We obtain the non-dimensional (primed variables) by using the characteristic scales given in Eq. (), which leads to 10∂vi′∂xi′=0,∂τij′∂xj′-∂P′∂xi′+Fi′=0,∂T′∂t′+vi′∂T′∂xi′=∂2T′∂xi′2+τij′ϵ′˙ij,ϵ′˙ij=12∂vi′∂xj′+∂vj′∂xi′=2-nτII′n-1exp⁡(nT′1+T′T0′)τij′, where Fi′ is now defined as Fi′=F‾1,0,-cot⁡(α) and F‾=ρgsin⁡(α)L‾/τ‾. The model parameters are the non-dimensional initial temperature T0′, the stress exponent n, the non-dimensional force F‾, the mean bed slope α, non-dimensional domain height Lz′, and the horizontal domain size Lx′ and Ly′ (Fig. ). We motivate the chosen characteristic scales by their usage in other studies of thermomechanical strain localisation . In the interest of a simple notation, we will omit the prime symbols on all non-dimensional variables in the remainder of the paper.

We consider a specific 1-D mathematical case in which all horizontal derivatives vanish (∂/∂x=∂/∂y=0). The only remaining shear stress component τxz and pressure P are determined by analytical integration and are constant in time considering a fixed domain. We assume that stresses vanish at the surface, and we set both horizontal and vertical basal velocity components to 0. We then integrate the 1-D mechanical equation in the vertical direction and substitute it into the temperature equation, which leads to 11∂T(z,t)∂t=∂2T(z,t)∂z2+2(1-n)F‾Lz(n+1)1-zLz(n+1)exp⁡(nT(z,t)1+T(z,t)T0),vx(z,t)=2(1-n)F‾Lzn∫0z1-zLznexp⁡(nT(z,t)1+T(z,t)T0)dz. Notably, the velocity and shear heating terms (Eq. ) are now a function only of temperature and thus of depth and time. To obtain a solution to the coupled system, one only needs to numerically solve for the temperature evolution profile, while the velocity can then be obtained diagnostically by a simple numerical integration.

We discretise the coupled thermomechanical Stokes equations (Eq. ) using the finite-difference method on a staggered Cartesian grid. Among many numerical methods currently used to solve partial differential equations, the finite-difference method is commonly used and has been successfully applied in solving a similar equation set relating to geophysical problems in geodynamics . The staggering of the grid provides second-order accuracy of the method , avoids oscillatory pressure modes and produces simple yet highly compact stencils. The different physical variables are located at different locations on the staggered grid. Pressure nodes and normal components of the strain rate tensor are located at the cell centres. Velocity components are located at the cell mid-faces (Fig. ), while shear stress components are located at the cell vertices in 2-D e.g.. The resulting algorithms are well suited for taking advantage of modern many-core parallel accelerators, such as graphical processing units (GPUs) . Efficient parallel solvers utilising modern hardware provide a viable solution to resolve the computationally challenging coupled thermomechanical full Stokes calculations in 3-D. The power-law viscous ice rheology (Eq. ) exhibits a non-linear dependence on both the temperature and the strain rate: 12η=ϵII˙1-nnexp⁡-T1+TT0, where ϵ˙II is the square root of the second invariant of the strain rate tensor ϵ˙II=1/2ϵ˙ijϵ˙ij. We regularise the strain rate and temperature-dependent viscosity η to prevent non-physical values for negligible strain rates, ηreg=1/(η-1+η0-1). We use a harmonic mean to obtain a naturally smooth transition to background viscosity values at negligible strain rate η0.

We define temperature on the cell centres within our staggered grid. We discretise the temperature equation's advection term using a first-order upwind scheme, doing the physical time integration using either an implicit backward Euler or a Crank–Nicolson scheme. To ensure that our numerical results are not confounded by numerical diffusion, the grid Peclet number must be smaller than the physical Peclet number. Limiting numerical diffusion is one motivation for using high numerical resolution in our computations.

We rely on a pseudo-transient (PT) continuation or relaxation method to solve the system of coupled non-linear partial differential equations (Eq. ) in an iterative and matrix-free way . To this end, we reformulate the thermomechanical Eq. () in a residual form: 13-∂vi∂xi=fp,∂τij∂xj-∂P∂xi+Fi=fvi,-∂T∂t-vi∂T∂xi+∂2T∂xi2+τijϵ˙ij=fT. The right-hand-side terms (fp,fvi,fT) are the non-linear continuity, momentum and temperature residuals, respectively, and quantify the magnitude of the imbalance of the corresponding equations.

We augment the steady-state equations with PT terms using the analogy of physical transient processes such as the bulk compressibility or the inertial terms within the momentum equations . This formulation enables us to integrate the equation forward in pseudo-time τ until we reach the steady state (i.e. the pseudo-time derivatives vanish). Relying on transient physics within the iterative process provides well-defined (maximal) iterative time step limiters. We reformulate Eq. (): 14-∂vi∂xi=∂P∂τp,∂τij∂xj-∂P∂xi+Fi=∂vi∂τvi,-∂T∂t-vi∂T∂xi+∂2T∂xi2+τijϵ˙ij=∂T∂τT, where we introduced the pseudo-time derivatives ∂/∂τ for the continuity (∂P/∂τp), the momentum (∂vi/∂τvi) and the temperature (∂T/∂τT) equation.

For every non-linear iteration k, we update the effective viscosity ηeff[k] in the logarithmic space by taking a fraction θη of the actual physical viscosity η[k] using the current strain rate and temperature solution fields and a fraction (1-θη) of the effective viscosity calculated in the previous iteration ηeff[k-1]: 15ηeff[k]=exp⁡θηln⁡η[k]+(1-θη)ln⁡ηeff[k-1]. We use the scalar θη(0≤θη≤1) to select the fraction of a given non-linear quantity, here the effective viscosity ηeff, to be updated each iteration. When θη=0, we would always use the initial guess, while for θη=1, we would take 100% of the current non-linear quantity. We usually define theta to be in the range of 10-2-10-1 in order to account for some time to fully relax the non-linear viscosity as the non-linear problem may not be sufficiently converged at the beginning of the iterations. This approach is in a way similar to an under-relaxation scheme and was successfully implemented in the ice-sheet model development by , for example.

The pseudo-time integration of Eq. () leads to the definition of pseudo-time steps Δτp,Δτvi and ΔτT for the continuity, momentum and temperature equations, respectively. Transient physical processes such as compressibility (continuity equation) or acceleration (momentum equation) dictate the maximal allowed explicit pseudo-time step to be utilised in the transient process. Using the largest stable steps allows one to minimise the iteration count required to reach the steady state: 16Δτp=2.1ndimηeffk(1+ηb)max⁡(ni),Δτvi=min⁡(Δxi)22.1ndimηeffk(1+ηb),ΔτT=2.1ndimmin⁡(Δxi)2+1Δt-1, where ndim is the number of dimensions, and Δxi and ni are the grid spacing and the number of grid points in the i direction (i=x in 1-D, x,z in 2-D and x,y,z in 3-D), respectively. The physical time step, Δt, advances the temperature in time. The pseudo-time step ΔτT is an explicit Courant–Friedrich–Lewy (CFL) time step that combines temperature advection and diffusion. Similarly, Δτvi is the explicit CFL time step for viscous flow, representing the diffusion of strain rates with viscosity as the diffusion coefficient. It is modified to account for the numerical equivalent of a bulk viscosity ηb. We choose Δτp to be the inverse of Δτvi to ensure that the pressure update is proportional to the effective viscosity, while the velocity update is sensitive to the inverse of the viscosity. This interdependence reduces the iterative method's sensitivity to the variations in the ice's viscosity.

During the iterative procedure, we allow for finite compressibility in the ice, ∂P/∂τp, while ensuring that the PT iterations eventually reach the incompressible solution. The relaxation of the incompressibility constraint is analogous to the penalisation of pressure pioneered by and subsequently extended by others. Compared to projection-type methods, it has the advantage that no pressure boundary condition is necessary that will lead to numerical boundary layers . We use the parameter ηb to balance the divergence-free formulation of strain rates in the normal stress component evaluation, wherein it is multiplied with the pressure residual fp. Thus, normal stress is given by τii=2η(ϵ˙ii+ηbfp). With convergence of the method, the pressure residual fp vanishes and the incompressible form of the normal stresses is recovered.

Combining the residual notation introduced in Eq. () with the pseudo-time derivatives in Eq. () leads to the update rules: 17P[k]=P[k-1]+ΔP[k],vi[k]=vi[k-1]+Δvi[k],T[k]=T[k-1]+ΔT[k], where the pressure, velocity and temperature iterative increments represent the current residual [k] multiplied by the pseudo-time step: 18ΔP[k]=Δτpfp[k],Δvi[k]=Δτvifvi[k],ΔT[k]=ΔτTfT[k].

The straightforward update rule (Eq. ) is based on a first-order scheme (∂/∂τ). In 1-D, it implies that one needs N2 iterations to converge to the stationary solution, where N stands for the total number of grid points. This behaviour arises because the time step limiter Δτvi implies a second-order dependence on the spatial derivatives for the strain rates. In contrast, a second-order scheme , ψ∂2/∂τ2+∂/∂τ invokes a wave-like transient physical process for the iterations. The main advantage is the scaling of the limiter as Δx instead of Δx2 in the explicit pseudo-transient time step definition. We can reformulate the velocity update as 19Δvi[k]=Δτvifvi[k]+1-νniΔvi[k-1], where ψ can be expanded to (1-ν/ni) and acts like a damping term on the momentum residual. A similar damping approach is used for elastic rheology in the FLAC geotechnical software in order to significantly reduce the number of iterations needed for the algorithm to converge. The optimal value of the introduced parameter ν is found to be in a range (1≤ν≤10), and it is usually problem-dependent. This approach was successfully implemented in recent PT developments by and . The iteration count increases with the numerical problem size for second-order PT solvers and scales close-to-ideal multi-grid implementations. However, the main advantage of the PT approach is its conciseness and the fact that only one additional read/write operation needs to be included – keeping additional memory transfers to the strict minimum.

Notably, the PT solution procedure leads to a two-way numerical coupling between temperature and deformation (mechanics), which enables us to recover an implicit solution to the entire system of non-linear partial differential equations. Besides the coupling terms, rheology is also treated implicitly; i.e. the shear viscosity η is always evaluated using the current physical temperature, T, and strain rate, ϵ˙II. Our method is fully local. At no point during the iterative procedure does one need to perform a global reduction, nor to access values that are not directly collocated. These considerations are crucial when designing a solution strategy that targets parallel hardware such as many-core GPU accelerators. We implemented the PT method in the MATLAB and CUDA C programming languages. Computations in CUDA C can be performed in both double- and single-precision arithmetic. The computations in CUDA C shown in the remainder of the paper were performed using double-precision arithmetic if not specified otherwise.

Our GPU algorithm development effort is motivated by the aim to resolve the coupled thermomechanical system of equations (Eqs. –) with high spatial and temporal accuracy in 3-D. To this end, we exploit the low-level intrinsic parallelism of shared memory devices, particularly targeting GPUs. A GPU is a massively parallel device originally devoted to rendering the colour values for pixels on a screen independently from one another whereby the latency can be masked by high throughput (i.e. compute as many jobs as possible in a reasonable time). A schematic representation (Fig. ) highlights the conceptual discrepancy between a GPU and CPU. On the GPU chip, most of the area is devoted to the arithmetic units, while on the CPU, a large area of the chip hosts scheduling and control microsystems.

The development of GPU-based solvers requires time to be devoted to the design of new algorithms that leverage the massively parallel potential of the current GPU architectures. Considerations such as limiting the memory transfers to the mandatory minimum, avoiding complex data layouts, preferring matrix-free solvers with low memory footprint and optimal parallel scalability instead of classical direct–iterative solver types are key in order to achieve optimal performance.

Our implementation does not rely on the CUDA unified virtual memory (UVM) features. UVM avoids the need to explicitly define data transfers between the host (CPU) and device (GPU) arrays but results in about 1 order of magnitude lower performance. We suspect the internal memory handling to be responsible for continuously synchronising the host and device memory, which is not needed in our case.

We rely on a distributed memory parallelisation using the message passing interface (MPI) library to overcome the on-device memory limitation inherent to modern GPUs and exploit supercomputers' computing power. Access to a large number of parallel processes enables us to tackle larger computational domains or to refine grid resolution. We rely on domain decomposition to split our global computational domain into local domains, each executing on a single GPU handled by an MPI process. Each local process has its boundary conditions defined by (a) physics if on the global boundary or (b) exchanged information from the neighbouring process in the case of internal boundaries. We use CUDA-aware non-blocking MPI messages to exchange the internal boundaries among neighbouring processes. CUDA awareness allows us to bypass explicit buffer copies on the host memory by directly exchanging GPU pointers, resulting in an enhanced workflow pipelining. Our algorithm implementation and solver require no global reduction. Thus, there is no need for global MPI communication, eliminating an important potential scaling bottleneck. Although the proposed iterative and matrix-free solver features a high locality and should scale by construction, the growing number of MPI processes may deprecate the parallel runtime performance by about 20 % owing to the increasing number of messages and overall machine occupancy . We address this limitation by overlapping MPI communication and the computation of the inner points of the local domains using streams, a native CUDA feature. CUDA streams allow one to assign asynchronous kernel execution and thus enable the overlap between communication and computation, resulting in optimal parallel efficiency.

We compare our numerical solutions obtained with the GPU-based PT method using a CUDA C implementation (FastICE) to the reference Elmer/Ice model. We report all the values in their non-dimensional form, and the horizontal axes are scaled with their aspect ratio. We impose a no-slip boundary condition on all velocity components at the base and prescribe free-slip boundary conditions on all lateral domain sides. We prescribe a stress-free upper boundary in the vertical direction.

In the 2-D configuration (Fig. ), the horizontal velocity component vanishes at the left and right boundary, vx=0, and thus the maximum velocity values in the horizontal direction are located in the middle of the slab. On the left side (x/Lx=0), the ice is pushed down (compression); thus, the vertical velocity values were negative. On the right side (x/Lx=1), the ice is pulled up (extension), and the vertical velocity values were positive. Our FastICE results agree well with the numerical solutions produced by Elmer/Ice. The numerical resolution of the Elmer/Ice model is 1001×275 grid points in the x and z directions (≈8.25×105 degrees of freedom – DOFs), while we employed 2047×511 grid points (≈3.13×106 DOFs) within our PT method. We use higher numerical grid resolution within FastICE to jointly verify agreement with Elmer/Ice and convergence. The fact that we obtain matching results when increasing grid resolution significantly suggests that we have sufficiently resolved the relevant physical processes, even at relatively low resolution. We report an exception to this trend in the 3-D case of Experiment 2. The PT method's efficiency enables simulations with a large number of grid points without affecting the runtime. The DOFs represent three variables in 2-D (vx,vz,P) and four variables in 3-D (vx,vy,vz,P) multiplied by the number of grid points involved.

We find good agreement between the two model solutions in the 3-D configuration as well (Fig. ). We employed a numerical grid resolution of 319×159×119 grid points in the x, y and z directions (≈2.41×107 DOFs) and used a numerical grid resolution of 61×61×21 (≈3.1×105 DOFs) in Elmer/Ice. Scaling our result to dimensional values (Table ) results in a maximal horizontal velocity (vx) of ≈105 m yr-1. The horizontal distance is 2 km in the x direction and 800 m in the y direction, and the ice thickness is 200 m. The box is inclined at 10∘.

We then consider the case in which ice is sliding at the base (ISMIP experiments C and D). We prescribe periodic boundary conditions at the lateral boundaries and apply a linear sliding law at the base. The top boundary remains stress-free in the vertical direction.

We performed the 2-D simulation of Experiment 2 (Fig. ) using a numerical grid resolution of 511×127 grid points (≈1.95×105 DOFs) for the FastICE solver and computed the Elmer/Ice solution using a numerical grid resolution of 241×120 (≈8.7×104 DOFs). We show both vx and vz velocity components at the slab's surface. The two models' results agree well.

We performed the 3-D simulation of Experiment 2 (Fig. ) using a numerical grid resolution of 63×63×21 (≈3.33×105 DOFs) for our FastICE solver and a numerical grid resolution of 61×61×21 (≈3.12×105 DOFs) in the Elmer/Ice model. In dimensional units, the maximum horizontal velocity (vx) corresponds to ≈16.4 m yr-1. The horizontal distance is 10 km in the x direction 10 km in the y direction, and the ice thickness is 1 km. The box is inclined at 0.1∘.

We find good agreement between the two numerical implementations. Since the flow is mainly oriented in the x direction, the vy velocity component is more than 2 orders of magnitude smaller than the vx velocity component. Numerical errors in vy are more apparent than in the leading velocity component vx. We report a 1 order of magnitude increase in the time to solution in Experiment 2 compared to the Experiment 1 configuration owing to the periodicity on the lateral boundaries.

We employ a matching numerical resolution between FastICE and Elmer/Ice in this particular benchmark case. Using higher resolution for FastICE results in minor discrepancy between the two solutions, suggesting that the resolution in Fig.  is insufficient to capture small-scale physical processes. We discuss this issue more in Sect.  where we test the convergence of the FastICE numerical implementation upon grid refinement.

We first verify that the 1-D, 2-D and 3-D model configurations from Experiment 3 produce identical results, assuming periodic boundary conditions on all lateral sides. In this case, all the variations in the x or y directions vanish (∂/∂x and ∂/∂y); thus, both the 2-D and 3-D models reduce to the 1-D problem. We employ a numerical grid resolution of 127×127×127 grid points in the x, y and z direction, 127×127 grid points in the x and z directions, and 127 grid points in the z direction for the 3-D, 2-D and 1-D problems, respectively.

We ensure that all results collapse onto the semi-analytical 1-D model solution (Sect. ), which we obtained by analytically integrating the velocity field and solving the decoupled thermal problem separately (Eq. ). From a computational perspective, we numerically solve Eq. () using a high spatial and temporal accuracy and therefore minimise the occurrence of numerical errors. We establish the 1-D reference solution for both the temperature and the velocity profile, solving Eq. () on a regular grid, reducing the physical time steps until we converge to a stable reference solution. Our reference simulation involves 4000 grid points and a non-dimensional time step of 5×105 (using a backward Euler time integration). We reach the total simulation time of 2.9×108 within 580 physical time steps.

We report overall good agreement of all model solutions (1-D, 2-D, 3-D and 1-D reference) at the three reported stages for this scenario (Fig. ). As expected from the 1-D model solution, temperature varies only as a function of time and depth, with the highest value obtained close to the base and for longer simulation times. Similarly, the velocity profile is equivalent to the 1-D profile, and the largest velocity value is located at the surface. We only report the horizontal velocity component vx for the 2-D and the 3-D models, since vy and vz feature negligible magnitudes. Thus, we only observe spatial variation in the vertical z direction. We report the non-dimensional temperature T (Fig. a) and horizontal velocity vx (Fig. b) fields for both the 3-D and the 2-D configurations compared at non-dimensional time 0.7×108, 1.4×108 and 1.9×108. The dimensional results from Experiment 3 correspond to a 300 m thick ice slab inclined at a 10∘ angle with an initial surface temperature of -10 ∘C. The maximum initial velocity for the isothermal ice slab corresponds to ≈486 m yr-1, while the maximum velocity just before the melting point is reached corresponds to 830 m yr-1. The comparison snapshot times are 1.6, 3.2 and 4.4 years.

The semi-analytical 1-D solution enables us to evaluate the influence of the numerical coupling method and time integration and to quantify when and why high spatial resolution is required in thermomechanical ice-flow simulations. We compare the 1-D semi-analytical reference solution (Eq. ) to the results obtained with the 1-D FastICE solver for three spatial numerical resolutions (nz=31, 95 and 201 grid points) at three non-dimensional times 1×108, 2×108 and 2.9×108 (Fig. ). The grey area in Fig.  highlights where the melting temperature is exceeded. Since our semi-analytical reference solution does not include phase transitions, we also neglect this component in the numerical results. During the early stages of the simulation, the thermomechanical coupling is still minor, and solutions at all resolution levels are in good agreement with one another and with the reference. The low-resolution solution starts to deviate from the reference (Fig. b) when the coupling becomes more pronounced close to the thermal runaway point . The high-spatial-resolution solution is satisfactory at all stages. We conclude that high spatial resolution is required to accurately capture the non-linear coupled behaviour in regimes close to the thermal runaway, which is seldom the case in the models reported in the literature.

Thermomechanical strain localisation may significantly impact the long-term evolution of a coupled system. A recent study by suggested that partial coupling may result in underestimating the thermomechanical localisation compared to the fully coupled approach, as reported in their Fig. 8. We compare three coupling methods (Fig. ): (1) a fully coupled implicit PT method, as described in the numerical section, whereby the viscosity and the shear heating term are implicitly determined by using the current guess; (2) an implicit numerically uncoupled mechanical and thermal model; and (3) an explicit numerically uncoupled mechanical and thermal model. The numerical time integration in physical time is performed using an implicit backward Euler method for (1) and (2) and a forward Euler explicit time integration method for (3). We utilise the identical non-dimensional time step for both the explicit and the implicit numerical time integration. We perform 580 time steps, reaching a simulation time of 2.9×108. We employ a vertical grid resolution of nz=201 grid points for all models. The chosen time step for the explicit integration of the heat diffusion equation is below the CFL stability condition given by Δz2/2.1 in 1-D, where Δz represents the grid spacing in a vertical direction.

Physically, the viscosity and shear heating terms are coupled and are a function of temperature and strain rates, but we update the viscosity and the shear heating term based on temperature values from the previous physical time step. Thus, the shear heating term can be considered a constant source term in the temperature evolution equation during the time step, leading to a semi-explicit rheology. We show the 1-D numerical solutions of (blue) the fully coupled method with a backward Euler (implicit) time integration and the two uncoupled methods with either (green) backward (implicit) or (red) forward (explicit) Euler time integration (Fig. ) and compare them to the 1-D reference model solution. Surprisingly, and in contrast to , we observe a good agreement between all methods, suggesting that the different coupling strategies capture the coupled flow physics with sufficient accuracy given high enough spatial and temporal resolution. However, for a longer-term evolution, the uncoupled approaches may predict lower temperature and velocity values than the fully coupled approach.

Boundary conditions corresponding to immobile regions in the computational domain may induce localisation of deformation and flow observed in locations such as shear margins, grounding zones or bedrock interactions. Dimensionality plays a key role in such configurations, causing the stress distribution to be variable among the considered directions.

We used the configuration in Experiment 3 to investigate the spatial variations in temperature and velocity distributions by defining no-slip conditions on the lateral boundaries for the mechanical problem and prescribing zero heat flux through those boundaries. We employ a numerical grid resolution of 511×255×127 grid points, 511×127 grid points and 201 grid points for the 3-D, 2-D and 1-D case, respectively. We prescribe a non-dimensional time step of 5×105. We perform 500 numerical time steps and reach a total non-dimensional simulation time of 2.5×108. We then compare the temperature T and horizontal velocity component vx at three times obtained with the 1-D, 2-D and 3-D FastICE solver to the reference solution (Fig. ). We use 1-D profiles for comparison, taken at location x=Lx/2 in the 2-D model and at location x=Lx/2 and y=Ly/2 in the 3-D model. We also report the temperature variation ΔT (Fig. a) and the horizontal velocity component vx (Fig. b) for both the 2-D and 3-D simulations. The melting temperature approximately corresponds to 0.35 of the temperature deviation. The reported results correspond to a 2.3-, 4.6- and 5.8-year evolution.

All three models start with identical initial conditions for the thermal problem, i.e. ΔT=0 throughout the entire ice slab. The difference between the models arises owing to different stress distributions in 1-D, 2-D or 3-D. For instance, the additional stress components inherent in 2-D and 3-D are of the same order of magnitude as the 1-D shear stress for the considered aspect ratio, reducing the horizontal velocity vx in the 2-D and 3-D models. This also impacts the shear heating term, reducing the source term in the temperature evolution equation. In the 1-D configuration, the unique shear stress tensor component is a function only of depth. On the other endmember, the 3-D configurations allow for a spatially more distributed stress state. They lower strain rates in this scenario and reduce the magnitude of shear heating in higher dimensions. The spatially heterogeneous temperature and strain rate fields in all directions require the utilisation of sufficiently high spatial numerical resolution in all directions in order to accurately resolve spontaneous localisation.

We did not consider phase transition in the previous experiments for the sake of model comparison and because the analytical solution excluded this process. The existence of a phase transition caps the temperature at the pressure melting point in regions with pronounced shear heating, as illustrated in 2-D in Fig. . The simulation represents the thermomechanically coupled Experiment 3 with no sliding and thermally impermeable walls (similar to Fig. ). Meltwater production consumes excess heat generated by shear heating. Thus, melting provides a physical mechanism that avoids thermal runaway in shear-heating-dominated zones in the ice. The experiment duration in dimensional units is 70 years, and the maximal temperature increase is 10 ∘C upon reaching the melting point.

In order to confirm the accuracy of the FastICE numerical implementation, we report truncation errors (L2 norms) upon numerical grid refinement. We consider both the 2-D and 3-D configurations of Experiment 2 for this convergence test. We vary the numerical grid resolution, keeping the relative grid step Δx and Δy (and Δz in 3-D) ratio. We utilise a high-resolution numerical simulation as a reference and perform three additional simulations in which we keep dividing the number of grid points in both the x and y (and z in 3-D) direction by a factor of 2. We report the L2 norms, 21||Perr||2=||Pref-Pcoarse||2,||vxerr||2=||vxref-vxcoarse||2, for both the pressure P and the horizontal downslope vx velocity component on a logarithmic plot for both the 2-D (Fig. a) and 3-D configurations (Fig. b). The FastICE numerical implementation converges with increasing numerical resolution, and we report linear fitting slopes of -1.19 for pressure and about -1.4 for horizontal the velocity component.

We additionally report the behaviour of the residuals' converge as a function of the non-linear iterations niternonlin for the FastICE GPU-based implementation (Fig. a). The reported convergence history stands for a 2-D configuration of Experiment 3 and a numerical grid resolution of 511×127 grid points. The optimal damping parameter used in this case is ν=2 (Eq. ). We further report the sensitivity of the accelerated PT scheme on the damping parameter ν (Fig. b). We show that selecting the optimal damping parameter (in the reported case ν=2) ensures a relatively low number of iterations to converge both the linear and non-linear thermomechanical problem.

We used two metrics to assess the performance of the developed FastICE PT algorithm: the effective memory throughput (MTPeff) and the wall time. We first compare the effective memory throughput of the vectorised MATLAB CPU implementation and the single-GPU CUDA C implementation. We employ double-precision (DP) floating-point arithmetic in CUDA C for fair comparison. Second, we employ the wall-time metric to compare the performance of our various implementations (MATLAB, CUDA C) and compare these to the time to solution of the Elmer/Ice solver.

We use two methods to solve the linear system in Elmer/Ice. In the 2-D experiments, we use a direct method, and in 3-D, we use an iterative method. The direct method used in 2-D relies on the UMFPACK routines to solve the linear system. To solve the 3-D experiments, we employ the available bi-conjugate gradient-stabilised method (BICGstab) with an ILU0 preconditioning. We employ the configuration in Experiment 1 for all the performance measurements. We use an Intel i7 4960HQ 2.6 GHz (Haswell) four-core CPU to benchmark all the CPU-based calculations. For simplicity, we only ran single-core CPU tests, staying away from any CPU parallelisation of the algorithms. Thus, our MATLAB or the Elmer/Ice single-core CPU results are not representative of the CPU hardware capabilities and are only reported for reference.

The FastICE PT solver relies on evaluating a finite-difference stencil. Each cell of the computational domain needs to access neighbouring values in order to approximate derivatives. These memory access operations are the performance bottleneck of the algorithm, making it memory-bounded. Thus, the algorithm's performance depends crucially on the memory transfer speed, and not the rate of the floating-point operations. Memory-bounded algorithms place additional pressure on modern many-core processors, since the current chip design tends toward large flop-to-byte ratios. Over the past years and decades, the memory bandwidth increase has been much slower compared to the increase in the rate of floating-point operations.

As shown by and , a relevant metric to assess the performance of memory-bounded algorithms is the effective memory throughput (MTPeff) (Eq. ). The MTPeff determines how efficiently data are transferred between the main memory and the arithmetic units and is inversely proportional to the execution time: 22MTPeff=(nxnynz)niternIOnp10243tntGBs-1, where (nxnynz) stands for the total number of grid points, niter is the total number of numerical iterations performed, np is the arithmetic precision (single – 4 bytes or double – 8 bytes), tnt is the wall time in seconds needed to compute the niter iterations and nIO is the performed number of memory accesses. It represents the minimum number of memory operations (read-and-write or read only) required to solve a given physical problem. For instance, in the mechanical Stokes solver for Experiment 1, we have to update (read-and-write) three arrays (vx,vz and P) at every iteration in 2-D and four arrays (vx,vy,vz and P) at every iteration in 3-D. Thus, the update of the mandatory arrays requires a minimum of six (eight) read-and-write operations in 2-D (3-D). One additional read-and-write is needed to resolve the non-linear viscosity; thus, nIO=10 in the 2-D case and nIO=12 in 3-D.

We report MTPeff values obtained with the FastICE algorithm for both the vectorised MATLAB (CPU) and the CUDA C (GPU) implementations in double-precision arithmetic (Fig. a). We also show the GPU performance using single-precision arithmetic (Fig. a – green diamonds). The results we obtain should be compared to the peak memory throughput value MTPpeak for the specific hardware used. The MTPpeak reports the memory transfer rates delivered only by performing memory copy operations with no computations. This value reflects the hardware performance limit and the maximal effective memory bandwidth. We measure MTPpeak values for the Intel i7 4960HQ CPU of 20 GB s-1 and of 260 GB s-1 for the Nvidia Titan X GPU. The single-core vectorised MATLAB CPU implementation achieves about 0.7 GB s-1, and the CUDA C implementation 16 GB s-1. Thus, the MATLAB single-core CPU implementation reaches 3.5 % of the (CPU) hardware peak value, and the CUDA C (GPU) implementation at about 6.15 % and 11 % of the (GPU) hardware peak value using double-precision and single-precision arithmetic, respectively. Further improvement of the GPU MTPeff values can be achieved by optimising the GPU code using more on-the-fly calculations and advanced kernel scheduling.

We investigate the wall time to solve one time step with the FastICE GPU solver for both the 2-D and the 3-D configurations (Fig. b). We found wall times of about 15 min to solve ≈2.4×107 DOFs with double-precision arithmetic and only 3 min when using single-precision arithmetic on a Nvidia Titan X (Maxwell) GPU. In future investigations, one may consider comparing wall times obtained by CPU algorithms fully enabling all cores of the CPU against wall times for GPUs within the same price and power consumption range.

The 3-D performance results obtained on various available Nvidia GPUs are summarised in Fig. . We performed all the calculations using double-precision arithmetic. We compare the MTPeff and wall-time values as functions of the DOFs. We tested GPUs from various price ranges and chip generations, targeting entry-level GPUs such as the Nvidia Quadro P1000 (Pascal), high-end gaming cards such as the Nvidia Titan Black (Kepler) or the Nvidia Titan X (Maxwell), and data-centre-class GPU accelerators such as the Nvidia Tesla V100 PCIe (Volta). The MATLAB implementation peak MTPeff values are about 0.46 GB s-1, the Quadro P1000 (Pascal) values about 4.3 GB s-1, the Titan Black (Kepler) 12.4 GB s-1, the Titan X (Maxwell) 16.7 GB s-1 and the Tesla V100 (Volta) 83.2 GB s-1. The MTPeff values directly impact the wall time, since the memory bandwidth is the bottleneck in FastICE. We solved a 3-D problem involving 511×255×127 grid points (6.6×107 DOFs) in about 1 h on the Titan Black GPU, 40 min on the Titan X GPU, and only 8 min on the Tesla V100 GPU. Notably, at this resolution, we employed about 4.5 GB of memory to solve the isothermal Stokes model. The results suggest that more recent GPUs such as the data-centre Tesla V100 (Volta) offer a more significant (order of magnitude higher) performance increase than entry-level GPU accelerators, such as the Quadro P1000.

We share the performance of the GPU-MPI implementation of FastICE to execute on distributed memory machines. We achieve a weak scaling parallel efficiency of 99 % on the 512 Nvidia K80 (Kepler) GPUs on the Xstream Cray CS-Storm GPU compute cluster at the Stanford Research Computing Facility. As our baseline, we use a non-MPI single-GPU calculation. We then repeat the experiment using 1 to 512 MPI processes (thus GPUs) and report the normalised execution time (Fig. ). The effective drop in parallel efficiency is only 1 % involving 1 to 512 MPI processes. We achieve this close-to-optimal parallel efficiency by overlapping MPI message communication and local domain stencil calculations. We specifically employ distinct CUDA streams in order to execute the communication and computation overlap asynchronously. We repeat a similar experiment on both the Volta node, an 8 Nvidia Tesla V100 32 GB (Nvlink Volta) GPU compute node (analogous to Nvidia's DGX-1 box), and the octopus supercomputer hosting 128 consumer electronics Nvidia Titan X (Maxwell) GPUs at the Swiss Geocomputing Centre, University of Lausanne, Switzerland. On the Volta node, we report a weak scaling parallel efficiency of 0.985 % for a single MPI process running at 0.99 % of the non-MPI reference. On the octopus supercomputer, we report a parallel efficiency of 95.5 % with an effective drop in parallel efficiency of only 2 % involving 1 to 128 MPI processes.