Finite-Budget Convergence and Chaos-Aware Stopping Criteria for Time-Parallel Methods
Time-parallel algorithms, such as Parareal, are well understood for linear problems, but their convergence analysis for nonlinear, chaotic systems remains limited. This paper introduces a theoretical framework for analysing time-decomposition methods as local contraction mappings with a finite iteration budget. We derive a finite-budget guarantee linking the initial error, convergence rate, and iteration count through a geometric outer--inner-ball condition. We apply this framework to Parareal, deriving explicit estimates for the convergence factor on nonlinear problems. A frozen-Jacobian expansion separates tableau mismatch from Jacobian drift; actual Runge--Kutta maps admit short-window contraction under uniform neighbourhood hypotheses. First-batch defects estimate the initial error, and sampled actual-map derivatives give practical budget predictions covering all six tested weighted-error crossings; guarantees require uniform bounds. For chaos-aware stopping, we introduce a \emph{proximity function} that weights the interface defects (the jumps between chunks) and bounds weighted trajectory error. Numerical experiments on the logistic, Lorenz-63, and Lorenz-96 systems assess early acceptance. With accurate coarse propagation, the chaotic tests typically take two or three fine batches, and accurate prefixes can extend beyond exact chunks. In an independent Lorenz-63 test with inadequate coarse propagation, Lyapunov stopping reduces energy-estimation error relative to coarse-only at lower modelled cost than standard stopping and meets the prescribed adequacy test. Its gain over the zero-correction first fine batch is not significant, however, and a less costly intermediate-step serial solver meets the accuracy limit. An independent Lorenz-96 test finds no clear energy-error improvement. Statistical fidelity is assessed separately from weighted acceptance.