On a New Class of BDF and IMEX Schemes for Parabolic Type Equations

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.