Assessing the robustness and scalability of the accelerated pseudo-transient method

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 (PT) iterative method and present 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 PT solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency of more than 96 % on 2197 GPUs in distributed-memory parallelisation weak-scaling benchmarks. The 2197 GPUs allow for unprecedented tera-scale solutions of 3D variable viscosity Stokes flow on 4995(3) grid cells involving over 1.2 trillion degrees of freedom (DoFs). 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 direct-iterative solvers based on central processing units (CPUs) in terms of wall time, even at relatively low spatial resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy thus has a great potential for future high-performance computing (HPC) applications, and for paving the road to exascale in the geosciences and beyond.

Automated summary

The article introduces an accelerated pseudo-transient iterative method for efficient handling of numerical algorithms in highly parallel hardware. The method is supported by theoretical predictions and is implemented on graphical processing units (GPUs), achieving high parallel efficiency and enabling tera-scale solutions.