Explicit third-order isochronous adaptive time-stepping for coupled first- and second-order transient systems

This paper presents a universal adaptive time-stepping framework based on the concept of Isochronous integration (i[jls-end-space/]-Integration). Unlike traditional practices that employ distinct algorithmic frameworks for first-order (e.g., parabolic diffusion) and second-order (e.g., hyperbolic wave propagation) transient systems, i[jls-end-space/]-Integration implies that the same (iso) computational structure is employed for both. Consequently, the present unified framework involves a single analysis technology that can be readily switched from one system to the other via a novel adaptation procedure. Built upon a third-order explicit multistep-multistage integration scheme, a key innovation of this work is the decoupling of estimator complexity from solver accuracy: by employing a streamlined low-order approximation strategy to guide high-order integration, the framework reduces computational overhead without compromising global convergence. This universality is particularly advantageous for coupled problems with mixed time-order derivatives, where conventional approaches typically require segregated formulations. Comprehensive validation studies demonstrate that the estimator achieves excellent agreement with exact local errors, maintaining an effectivity index near unity even under randomized time-step scenarios. Numerical benchmarks, including thermal-structural coupling and phase-field solidification, confirm that the proposed approach yields significant improvements in computational efficiency while ensuring robust accuracy for complex multi-disciplinary simulations.