STABILITY OF GENERALIZED BDF2 AND AM2 METHODS

The main part of this thesis is to develop a unified approach to the uniform-in-time energy stability of a special class of linear two-step numerical methods when applied to some prototype linear dissipative problems. Also, some classical stability properties of these multi-step methods are analyzed, including A-stability(linear stability), G-stability, stability region, and decay rate. The aim is to construct better multi-step methods based on the analysis of these classical properties. Starting from recalling several classical linear multi-step methods, including the classical Adams-Bashforth(AM) methods and Backward differentiation formula(BDF) methods, we borrow the idea of Talor’s expansion based on different time point 𝑡 ∗ in order to construct the generalized two-step methods. After that, considering the efficiency and stability of numerical methods, several stability properties are recalled. It is natural to explore the stability of the generalized two-step methods, especially the generalized second-order Adams-Bashforth(AM2) method and the generalized second-order Backward differentiation formula(BDF) method. The former is derived by the expansion centered on the mid-point of the current and future time grid, while the latter is centered on the future time grid. In terms of the formulas, the generalized AM2 and the generalized BDF2 method are obtained by introducing a free parameter in their classical version. Therefore, we expect the generalized method can have better properties than the classical one, i.e., a lower local error and better stability. As a result, we find that the generalized AM2 method and the generalized BDF2 method are sometimes more stable than their classical version, i.e., with a larger stability region and larger decay rate. Naturally, we expect to make full advantage of these generalized two-step methods in some problems involving long-time behavior such as the coarsening process for the Cahn-Hilliard and several phase-field thin-film models. Lots of past research about the long-time stability issue of numerical schemes motivates us to explore whether the generalized AM2 method and the generalized BDF2 method can inherit the uniform-in-time energy bound in some linear dissipative problems. We eventually have some theoretical results about the uniform-in-time energy stability of the generalized AM2 and the generalized BDF2 methods in some prototype linear dissipative systems with a symmetric positive definite operator and a mild anti-symmetric operator. Additionally, some numerical results are given to show the advantages of these methods in accuracy and stability. On the other hand, we also try to explore the relationship between several stability properties, especially A-stability and uniform-in-time energy stability. It shows that in some cases, A-stability has an equivalent relationship with uniform-in-time energy stability in linear two-step convergent methods. And it can be directly used in some simple linear dissipative models to obtain the unconditionally uniform-in-time energy stability

本论文的主要内容是对一类特殊的线性两步数值方法在应用于一些原型线性 耗散问题时的均匀时间内的能量稳定性进行了一些统一的研究。同时，还分析了 这些多步骤方法的一些经典稳定性特性，包括 A-稳定性（线性稳定性）、G-稳定性、 稳定区域和衰减率。目的是在分析这些经典性质的基础上构建更好的多步骤方法。 文章正文从回顾几个经典的线性多步方法开始，包括经典的 Adams-Bashforth(AM) 方法和 Backward differentiation formula(BDF) 方法，我们借用基于不同时间点 𝑡 ∗ 的 泰勒展式思想来构造一般的两步方法。之后，考虑到数值方法的效率和稳定性，我 们回顾了几个稳定性性质。探讨一般两步法的稳定性是很自然的，尤其是广义二 阶 Adams-Bashforth(AM2) 方法和广义二阶向后差分公式 (BDF2)。前者是通过以当 前和未来时间网格的中点为中心的扩展得出的，而后者是以未来时间网格为中心 的。就公式而言，广义的 AM2 和广义的 BDF2 方法是通过在其经典版本中引入一 个自由参数而得到的。因此，我们希望广义的方法能比经典的方法有更好的特性， 即更低的局部误差和更稳定地性质。结果是，我们发现广义的 AM2 方法和广义的 BDF2 方法有时比它们的经典版本更稳定，即具有更大的稳定区域和更大的衰减 率。当然，我们期望在一些涉及长时间行为的问题中充分利用这些广义两步法，如 Cahn-Hilliard 和几个相场薄膜模型的粗粒化过程。过去对数值方案的长时间稳定 性问题的大量研究，促使我们探索广义 AM2 方法和广义 BDF2 方法是否能在一些 线性耗散问题中继承均匀时间内的能量约束。我们最终得到了一些关于广义 AM2 和广义 BDF2 方法在一些具有对称正定算子和特殊反对称算子的线性耗散系统中 的关于时间一致的能量稳定性的理论结果。此外，我们还给出了一些数值结果以 显示这些方法在精度和稳定性方面的优势。另一方面，我们还试图探索几个稳定 性性质之间的关系，特别是线性稳定性和关于时间一致的能量稳定性。结果表明， 在某些情况下，线性稳定性与收敛的线性两步方法中的时间一致能量稳定性具有 等价关系。而且它可以直接用于一些简单的线性耗散模型中去得到无条件的时间 一致能量稳定性。

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