On a new class of BDF and IMEX schemes for parabolic type equations

TL;DR

This paper introduces a new class of BDF and IMEX schemes for parabolic equations that enable larger time steps at higher order for stiff problems by using Taylor expansions with a tunable parameter.

Contribution

The paper develops a novel class of BDF and IMEX schemes based on Taylor expansions with a tunable parameter, improving stability and efficiency for stiff parabolic equations.

Findings

New schemes allow larger time steps at higher order for stiff problems.

Rigorous stability and error analysis conducted for second- to fourth-order schemes.

Numerical examples validate the theoretical advantages of the new schemes.

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 (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.