When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new schemes, with a suitable $\beta$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.

翻译：在应用经典多步格式求解微分方程时，常面临高阶格式需采用更小时步长的困境，这使得高阶格式难以用于刚性问题。本文基于在时间$t^{n+\beta}$处（其中$\beta > 1$为可调参数）的泰勒展开，构造了一类适用于抛物型方程的新型BDF与隐式-显式（IMEX）格式。通过选取合适的$\beta$，这些新格式对刚性问题允许比常规高阶格式更大的时步步长。针对抛物型方程，我们确定了新格式（二阶至四阶）的显式统一乘子，并采用能量论证严格开展了稳定性与误差分析。同时，我们通过大量数值算例验证了理论结果。