For implicit–explicit multistep schemes, using a suitable form of Dahlquist’s test equation, we introduce an analogue to the A(θ)-stability concept, valid for implicit methods, and formulate a stability criterion in terms of an auxiliary function that plays a key role in our analysis. Furthermore, for implicit–explicit backward difference formula methods, we either evaluate the auxiliary function or establish very good estimates of it; as a result, we derive a sharp or very good, respectively, unconditional stability condition, the analogue of the determination of the exact angle θ for implicit methods or of a good approximation thereof. A comparison with the corresponding necessary stability condition provides evidence of the quality of the sufficient stability condition. In addition, we verify our analysis with results of a series of numerical experiments.

中文翻译：

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隐式-显式 BDF 方法的 A$(\vartheta)$-稳定性概念的类比

对于隐式-显式多步方案，使用合适形式的 Dahlquist 测试方程，我们引入了 A(θ)-稳定性概念的类似物，对隐式方法有效，并根据起关键作用的辅助函数制定稳定性标准在我们的分析中的作用。此外，对于隐式-显式后向差分公式方法，我们要么评估辅助函数，要么对其建立非常好的估计；因此，我们分别推导出了一个尖锐的或非常好的无条件稳定性条件，类似于确定隐式方法的精确角度 θ 或其良好近似。与相应的必要稳定性条件的比较提供了充分稳定性条件质量的证据。此外，