Assessing the robustness and scalability of the accelerated pseudo-transient method towards exascale computing

Abstract

Abstract. The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient iterative method and present here a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of pseudo-transient solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency over 96 % on 2197 GPUs in distributed memory parallelisation weak scaling benchmarks. 2197 GPUs allow for unprecedented terascale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom. We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity, and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform CPU-based direct-iterative solvers in terms of wall-time even at relatively low resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy has thus a great potential for future high-performance computing applications, and for paving the road to exascale in the geosciences and beyond.

Control of nonlinear bulk deformation and large shear strain on first-order phase transformation kinetics

“…To solve the coupled system ( 33)-(41), we employ the accelerated pseudotransient (PT) method [25,27]. In accordance with this method, the steadystate solution to the mechanical problem is obtained as a limit of a transient physical process.…”

“…The values of these parameters can be calibrated to achieve fastest convergence. For linear equations the optimal values for G and θ can be determined from a dispersion analysis of the governing equations as reported in [27]. In the linear case, the number of iterations Article Title required for convergence grows linearly with increasing grid resolution.…”

“…The diffusion-and interface-controlled regimes have been studied analytically by Truskinovsky (1994) [16]. Here, we expand on the work of Truskinovsky (1994) [16] and consider five additional aspects: (1) We consider a phase transformation in a system with two chemical components, (2) we add the contribution of pressure-dependent mechanical mixing to the Gibbs energy of the system [13], (3) we apply a geologically relevant nonlinear equation of state in which the compressibility is a function of pressure [22][23][24], (4) we consider advection and large finite shear strains and (5) we solve the governing equations numerically in 1D and 2D employing a pseudo-transient finite difference method [25][26][27].…”

Spontaneous Formation of an Internal Shear Band in Ice Flowing over Topographically Variable Bedrocks

“…We include a low-viscosity phase on top of the ice to mimic the presence of air, which allows ice thickness to change spatially and temporally. We capture the depth-dependent thermo-mechanical ice deformation implementing an incompressible viscous Stokes solver using the time-dependent implicit pseudotransient methods (Räss et al, 2022) and the finite difference discretization developed by Räss et al (2020). To prescribe the basal ice-bedrock boundary condition, we implement the Immersed Boundary Method, a fictitious domain method that allows to treat fluid and structural domains separately (Peskin, 1972(Peskin, , 2002.…”

Performance portable ice-sheet modeling with MALI

“…FastICE is introduced in [58] as a parallel, GPU-accelerated full Stokes solver developed in CUDA, which utilizes a matrix-free method with pseudo-transient continuation. This is extended to a portable framework written in Julia in [59] and a parallel efficiency over 96% on 2197 GPUs is achieved.…”