Second-order unconditionally stable schemes of linear multi-step methods, and their equivalent single-step methods, are developed in this paper. The parameters of the linear two-, three-, and four-step methods are determined for optimal accuracy, unconditional stability and tunable algorithmic dissipation. The linear three- and four-step schemes are presented for the first time. As an alternative, corresponding single-step methods, spectrally equivalent to the multi-step ones, are developed by introducing the required intermediate variables. Their formulations are equivalent to that of the corresponding multi-step methods; their use is more convenient, owing to being self-starting. Compared with existing second-order methods, the proposed ones, especially the linear four-step method and its alternative single-step one, show higher accuracy for a given degree of algorithmic dissipation. The accuracy advantage and other properties of the newly developed schemes are demonstrated by several illustrative examples.

中文翻译：

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线性多步和等效单步积分方法的改进二阶无条件稳定方案

本文开发了线性多步方法的二阶无条件稳定方案及其等效的单步方法。确定线性两步、三步和四步方法的参数以获得最佳精度、无条件稳定性和可调算法耗散。首次提出了线性三步和四步方案。作为替代方案，通过引入所需的中间变量来开发相应的单步方法，在光谱上等效于多步方法。它们的公式与相应的多步方法相同；由于是自启动，它们的使用更加方便。与现有的二阶方法相比，所提出的方法，尤其是线性四步法及其替代的单步法，对于给定程度的算法耗散显示更高的准确性。新开发的方案的准确性优势和其他特性通过几个说明性示例得到了证明。