Maximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolic equations

TL;DR

This paper provides optimal error estimates for implicit-explicit BDF methods applied to nonlinear parabolic equations, extending the analysis to complex Banach spaces and various boundary value problems.

Contribution

It introduces a novel analysis framework for implicit-explicit BDF methods on nonlinear parabolic equations with time-dependent operators in Banach spaces.

Findings

Optimal order a priori error estimates established

Applicability demonstrated on four different boundary value problems

Conditions on non-self-adjointness of operators are sharp

Abstract

We establish optimal order a priori error estimates for implicit-explicit BDF methods for abstract semilinear parabolic equations with time-dependent operators in a complex Banach space settings, under a sharp condition on the non-self-adjointness of the linear operator. Our approach relies on the discrete maximal parabolic regularity of implicit BDF schemes for autonomous linear parabolic equations, recently established in [20], and on ideas from [7]. We illustrate the applicability of our results to four initial and boundary value problems, namely two for second order, one for fractional order, and one for fourth order, namely the Cahn-Hilliard, parabolic equations.