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<span id="GuEtAl2024b">H. Gu, M. Cai, and J. Li, “Crank-Nicolson-type iterative decoupled algorithms for Biot’s consolidation model using total pressure,” arXiv:2409.18391v1 [math.NA], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2409.18391v1" target="_blank">http://arxiv.org/abs/2409.18391v1</a></span>
In this work, we develop Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot’s consolidation model using total pressure. We begin by constructing an equivalent fully implicit coupled algorithm using the standard Crank-Nicolson method for the three-field formulation of Biot’s model. Employing an iterative decoupled scheme to decompose the resulting coupled system, we derive two distinctive forms of Crank-Nicolson-type iterative decoupled algorithms based on the order of temporal computation and iteration: a time-stepping iterative decoupled algorithm and a global-in-time iterative decoupled algorithm. Notably, the proposed global-in-time algorithm supports a partially parallel-in-time feature. Capitalizing on the convergence properties of the iterative decoupled scheme, both algorithms exhibit second-order time accuracy and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.
<span id="Hope-CollinsEtAl2024">J. Hope-Collins, A. Hamdan, W. Bauer, L. Mitchell, and C. Cotter, “asQ: parallel-in-time finite element simulations using ParaDiag for geoscientific models and beyond,” arXiv:2409.18792v1 [math.NA], 2024 [Online]. Available at: <a href="http://arxiv.org/abs/2409.18792v1" target="_blank">http://arxiv.org/abs/2409.18792v1</a></span>
Modern high performance computers are massively parallel; for many PDE applications spatial parallelism saturates long before the computer’s capability is reached. Parallel-in-time methods enable further speedup beyond spatial saturation by solving multiple timesteps simultaneously to expose additional parallelism. ParaDiag is a particular approach to parallel-in-time based on preconditioning the simultaneous timestep system with a perturbation that allows block diagonalisation via a Fourier transform in time. In this article, we introduce asQ, a new library for implementing ParaDiag parallel-in-time methods, with a focus on applications in the geosciences, especially weather and climate. asQ is built on Firedrake, a library for the automated solution of finite element models, and the PETSc library of scalable linear and nonlinear solvers. This enables asQ to build ParaDiag solvers for general finite element models and provide a range of solution strategies, making testing a wide array of problems straightforward. We use a quasi-Newton formulation that encompasses a range of ParaDiag methods, and expose building blocks for constructing more complex methods. The performance and flexibility of asQ is demonstrated on a hierarchy of linear and nonlinear atmospheric flow models. We show that ParaDiag can offer promising speedups and that asQ is a productive testbed for further developing these methods.
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