An adaptive single-step CTI-Bootstrapping temporal framework for the incompressible Navier-Stokes equations

This paper presents an adaptive, single-step Complex-Time Integration (CTI) Bootstrapping framework for the incompressible Navier-Stokes equations. Conventional explicit artificial-compressibility solvers are often limited by acoustic-CFL stability constraints, while history-dependent explicit extrapolations make robust variable-step integration cumbersome. To address these limitations, we evaluate spatial operators over a complex contour, extracting high-order temporal derivatives without recursive algebraic differentiation. Synthesizing these derivatives with a local Taylor prediction decouples temporal advancement from historical states, enabling step-size modulation. To eliminate the divergence defect, a hierarchical Bootstrapping sequence is applied. Crucially, applying a discrete projection algebraically reduces the complex continuity equation to a simple scalar update, avoiding global Poisson solvers. Governed by a hierarchy-embedded error estimator, the framework autonomously scales the integration step. Numerical experiments on staggered MAC grids confirm formal asymptotic convergence and algorithmic robustness under randomized step-sizes. The resulting framework provides a high-order, Poisson-free time-integration strategy, while its complex-contour residual evaluations introduce a non-negligible serial overhead that is explicitly quantified through residual-evaluation and CPU-cost accounting.