Accelerated pseudo-transient method for elastic, viscoelastic, and coupled hydro-mechanical problems with applications

Abstract. The Accelerated Pseudo-Transient (APT) method is a matrix-free approach used to solve partial differential equations (PDEs), characterized by its reliance on local operations, which makes it highly suitable for parallelization. With the advent of the memory-wall phenomenon around 2005, where memory access speed overtook floating-point operations as the bottleneck in high-performance computing, the APT method has gained prominence as a powerful tool for tackling various PDEs in geosciences. Recent advancements have demonstrated the APT method's computational efficiency, particularly when applied to quasi-static nonlinear problems using Graphical Processing Units (GPUs). This manuscript presents a comprehensive analysis of the APT method, focusing on its application to quasi-static elastic, viscoelastic, and coupled hydro-mechanical problems, specifically those governed by quasi-static Biot's poroelastic equations, across 1D, 2D, and 3D domains. We systematically investigate the optimal numerical parameters required to achieve rapid convergence, offering valuable insights into the method's applicability and efficiency for a range of physical models. Our findings are validated against analytical solutions, underscoring the robustness and accuracy of the APT method in both homogeneous and heterogeneous media. We explore the influence of boundary conditions, non-linearities, and coupling on the optimal convergence parameters, highlighting the method's adaptability in addressing complex and realistic scenarios. To demonstrate the flexibility of the APT method, we apply it to the nonlinear mechanical problem of strain localization using a poro-elasto-viscoplastic rheological model, achieving extremely high resolutions – 10,000² voxels in 2D and 512³ voxels in 3D – that, to our knowledge, have not been previously explored for such models. Our study contributes significantly to the field by providing a robust framework for the effective implementation of the APT method in solving challenging geophysical problems. Importantly, the results presented in this paper are fully reproducible, with Matlab, symbolic Maple scripts, and CUDA C codes made available in a permanent repository.