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

The development of highly efficient, robust, and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. In this work, we address this challenge with the accelerated pseudo-transient iterative method. This method is motivated by the physical analogy between numerical iterations and transient processes converging to a steady state.

We analytically determine optimal iteration parameters for a variety of basic physical processes such as diffusion, diffusion-reaction and non-inertial viscous fluid flow featuring Maxwell viscoelastic rheology. We further confirm the validity of theoretical predictions with numerical experiments.

We provide an efficient numerical implementation of various 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 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.

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.