Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part I: Stability

Abstract

Explicit exponential integrators based on general linear methods are studied for the time discretization of quasi-linear parabolic initial-boundary value problems. Compared to other exponential integrators encountering rather severe order reductions, in general, the considered class of exponential general linear methods provides the possibility to construct schemes that retain higherorder accuracy in time when applied to quasi-linear parabolic problems. Employing an abstract framework, the considered problems take the form of initial value problems on Banach spaces: u (t) = Q(u(t))u(t), t ∈ (0, T ), u(0) given. A fundamental requirement for the stability and error analysis is that the domains of the defining sectorial operatorsThe scope of applications in particular includes quasi-linear parabolic evolution equations subject to Dirichlet boundary conditions. The work is divided into two parts. In Part I, stability bounds in the norms of certain intermediate spaces between the domain D and the underlying Banach space X are deduced. In view of practical applications, the stability estimates are stated for variable time stepsizes, under mild restrictions on the ratios of subsequent stepsizes. The stability results provide a basic ingredient for the convergence analysis given in Part II. Introduction.Our present work on efficient time integration methods for quasi-linear parabolic problems comprises two parts. In this first part, we introduce the considered class of variable stepsize explicit exponential general linear methods and study their stability behavior. A second part shall be concerned with the convergence analysis of these exponential integrators.Scope of applications. Quasi-linear parabolic initial-boundary value problems typically arise in the modeling of minimal surfaces and mean curvature flow, in the study of fluids in porous media and sharp fronts in polymers, and for the description of thin fluid films and diffusion processes with state-dependent diffusivity; see, for instance, [3,4,12,14,15].Analytical framework. Employing the abstract framework of sectorial operators and analytic semigroups (see [2,19,22,25]), we may cast a quasi-linear parabolic

Higher-Order Exponential Integrators for Quasi-Linear Parabolic Problems. Part II: Convergence

“…. , s}) 5) are satisfied for certain Q, P ∈ N. In accordance with [6], we call Q ∈ N the stage order and P ∈ N the quadrature order of the method, see also [3]. Evidently, the following identity holds…”

“…The needed stability bounds, obtained under mild restrictions on the ratios of subsequent time stepsizes, have been deduced in the recent work [5]. The core of the present work is devoted to the derivation of suitable local and global error representations.…”

“…Quasi-linear parabolic initial-boundary value problems arise in the modelling of minimal surfaces and mean curvature flow, in the study of fluids in porous media and sharp fronts in polymers, and for the description of thin fluid films and diffusion processes with state-dependent diffusivity, see [5] and references given therein.…”

“…In the following, we recall the general format of exponential general linear methods and introduce auxiliary abbreviations. Additional details are given in [5].…”

“…In this work, we are concerned with the derivation of a convergence result for explicit exponential general linear methods of the form (1.3) applied to quasi-linear parabolic problems (1.1). We point out that, even though it is possible to sustain a general approach based on suitable local and global error representations, stability bounds, estimates for the defects, and the application of Gronwall-type inequalities, as used for instance in [6], the convergence analysis of quasi-linear parabolic problems is significantly more involved than the treatment of the semi-linear case, see also the discussion in [5]. In particular, it is essential to prove that the time-discrete solution is Hölder-continuous in a discrete sense, in analogy to the Hölder-continuity of the exact solution.…”