This work aims to introduce a heuristic timestep-adaptive algorithm for Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems where the flow is dominated by the pressure. In such scenarios, many time-adaptive algorithms based on the interplay of implicit and explicit time schemes fail to capture the fast transient dynamics of pressure fields. We present an algorithm that relies on a temporal error estimator using Backward Differentiation Formulae (BDF$k$) of order $k=2,3$. Specifically, we demonstrate that the implicit BDF$3$ solution can be well approximated by applying a single Newton-type nonlinear solver correction to the implicit BDF$2$ solution. The difference between these solutions determines our adaptive temporal error estimator. The effectiveness of our approach is confirmed by numerical experiments conducted on a backward-facing step flow CFD test case with Reynolds number $300$ and on a two-dimensional haemodynamics FSI benchmark.

翻译：本研究旨在为计算流体动力学（CFD）及流固耦合（FSI）问题中流动受压力主导的情形，提出一种启发式时间步长自适应算法。在此类场景下，许多基于隐式与显式时间格式相互作用的传统时间自适应算法难以捕捉压力场的快速瞬态动力学特征。本文提出一种算法，其依赖于采用$k=2,3$阶向后微分公式（BDF$k$）的时间误差估计器。具体而言，我们证明了通过对隐式BDF$2$解施加一次牛顿型非线性求解器修正，即可良好逼近隐式BDF$3$解。这两种解之间的差异构成了我们自适应时间误差估计器的基础。通过在雷诺数为$300$的后向台阶流动CFD算例以及一个二维血流动力学FSI基准测试上进行数值实验，验证了本方法的有效性。