We investigate the stability of two families of three-level two-step schemes that extend the classical second order BDF (BDF2) and second order Adams-Moulton (AM2) schemes. For a free parameter restricted to an appropriate range that covers the classical case, we show that both the generalized BDF2 and the generalized AM2 schemes are A-stable. We also introduce the concept of uniform-in-time stability which characterizes a scheme's ability to inherit the uniform boundedness over all time of the solution of damped and forced equation with the force uniformly bounded in time. We then demonstrate that A-stability and uniform-in-time stability are equivalent for three-level two-step schemes. Next, these two families of schemes are utilized to construct efficient and unconditionally stable IMEX schemes for systems that involve a damping term, a skew symmetric term, and a forcing term. These novel IMEX schemes are shown to be uniform-in-time energy stable in the sense that the norm of any numerical solution is bounded uniformly over all time, provided that the forcing term is uniformly bounded time, the skew symmetric term is dominated by the dissipative term, together with a mild time-step restriction. Numerical experiments verify our theoretical results. They also indicate that the generalized schemes could be more accurate and/or more stable than the classical ones for suitable choice of the parameter.

翻译：本文研究了两类扩展经典二阶BDF（BDF2）和二阶Adams-Moulton（AM2）格式的三层两步格式的稳定性。当自由参数限制在包含经典情形的适当范围内时，我们证明广义BDF2格式与广义AM2格式均是A-稳定的。同时，我们引入了时间一致稳定性的概念，该性质刻画了格式在驱动力随时间一致有界条件下，继承带阻尼项与强迫项方程的解在全时间范围内一致有界性的能力。随后我们证明：对于三层两步格式，A-稳定性与时间一致稳定性是等价的。进一步地，利用这两类格式构建了适用于含阻尼项、斜对称项及强迫项系统的高效无条件稳定IMEX格式。研究表明：当强迫项随时间一致有界、斜对称项受耗散项主导，并辅以温和的时间步长限制条件时，这些新型IMEX格式具有时间一致能量稳定性，即任意数值解的范数在全时间范围内一致有界。数值实验验证了理论结果，同时表明：通过选择合适的参数，广义格式可能比经典格式具有更高的精度和/或更好的稳定性。