Maximum norm analysis of implicit–explicit backward difference formulae for nonlinear parabolic equations

Abstract

We establish optimal order a priori error estimates for implicit–explicit backward difference formula (BDF) methods for abstract semilinear parabolic equations with time-dependent operators in a complex Banach space setting, 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 Kovács, Li & Lubich (2016, A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal., 54, 3600–3624), and on ideas from Akrivis, Li & Lubich (2017, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp.). We illustrate the applicability of our results to four initial and boundary value problems, namely two of second order, one of fractional order and one of fourth order, that is, the Cahn–Hilliard parabolic equations.