Pipelined Space-Time Krylov Method with Preconditioning: A Parallel-in-Time Algorithm for Biot’s Quasi-Static Poroelasticity

This paper presents a preconditioner-based parallel-in-time (PinT) method for solving the quasi-static Biot’s consolidation model in poroelasticity, a problem characterized by stiff coupling and saddle-point structures. To address the computational challenges of the resulting large-scale linear systems, we design two physics-based Schur-complement approximation preconditioners that ensure robust Krylov convergence. Crucially, the method achieves a pipelined space-time architecture by introducing an inverted time-stepping mechanism: Instead of sequential time marching, time steps are traversed in the inner loop, while the outer loop applies an iterative solve across the entire space-time trajectory. This structure relaxes the strict dependency on fully converged solutions at each time step, enabling approximate solutions to be iteratively refined in parallel. Implemented as a pipelined wavefront scheme with strictly nearest-neighbor communication, the algorithm achieves strong scalability. Algorithmic verification conducted on systems with up to 200 thousand degrees of freedom demonstrates stable convergence and sustained strong scaling with up to 128 cores. The proposed approach maintains the accuracy of the underlying finite element discretization while alleviating the “time bottleneck,” making it highly effective for large-scale, long-duration poroelastic simulations.