SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024.
Abstract. 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 schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which 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.

中文翻译：

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抛物型方程一类新的BDF和IMEX方案


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1609-1637 页，2024 年 8 月。

抽象的。当应用经典的多步格式求解微分方程时，人们经常面临这样的困境：高阶格式需要更小的时间步长，这使得使用高阶格式解决刚性问题是不切实际的。我们在本文中构建了一类新的 BDF 和基于时间 [math] 泰勒展开的抛物型方程的隐式-显式方案，其中 [math] 是可调参数。这些新方案具有合适的[数学]，与通常的高阶方案相比，对于刚性问题允许在更高阶上使用更大的时间步长。对于抛物型方程，我们为新的二阶到四阶格式确定了一个显式一致乘子，并使用能量参数进行严格的稳定性和误差分析。我们还提供了大量的数值例子来验证我们的发现。