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

Abstract

Abstract. In this work, the convergence analysis of explicit exponential time integrators based on general linear methods for quasi-linear parabolic initial-boundary value problems is pursued. Compared to other types of 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 higher-order accuracy in time when applied to quasi-linear parabolic problems. In view of practical applications, the case of variable time stepsizes is incorporated.The convergence analysis is based upon two fundamental ingredients. 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. In conjunction with the stability bounds, a convergence result is established. 1. Introduction. In the present work, we proceed our convergence analysis of explicit exponential integrators based on general linear methods [2] for quasi-linear parabolic problems. The considered class of time integration methods combines the benefits of exponential Runge-Kutta and exponential Adams-Bashforth methods and permits to construct explicit higher-order schemes that possess favorable stability properties for parabolic evolution equations. In view of practical applications, our investigations include the case of variable time stepsizes. In the first part [5], we have introduced the considered class of time integration methods, and we have proven stability in certain norms, under mild restrictions on the ratios of subsequent time stepsizes. This second part is devoted to the derivation of suitable local and global error representations and the resulting convergence estimates.Quasi-linear parabolic problems. 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 accordance with [5], we cast a quasi-linear parabolic initial-boundary value problem into the form of an initial value problem on a Banach space (X, · X )

One-stage explicit trigonometric integrators for effectively solving quasilinear wave equations

“…is seen from the second formula in (12) and (20). In a similar way, by the fourth formula in (12) it arrives that…”

“…which are obtained by considering the trigonometric scheme (11) and its symmetry. The same relations hold for v.…”

“…In contrast to the analysis in [10], we do not use a modified discrete energy in this paper and just take the simple and normal energy technique, which is widely used in the numerical analysis of partial differential equations (see, e.g. [1,5,6,9,11,12,16,19,22,27]). The paper is displayed as follows.…”

A high-order exponential integrator for nonlinear parabolic equations with nonsmooth initial data

“…This has been proved for most time discretization methods, including Runge-Kutta methods [4], implicit A(α)-stable multistep methods [20], implicit-explicit BDF methods [1,2], splitting methods [5,6,10] and several types of exponential integrators [3,11,12,25]. Extension to quasi-linear parabolic problems has also been done; see [7,8,9,13,14,21]. The error estimates presented in these articles do not apply to nonsmooth initial data.…”

Trigonometric integrators for quasilinear wave equations

“…We mention that recently in [14,15] explicit exponential integrators for quasilinear parabolic problems in Banach spaces were considered. Analysis of quasilinear parabolic equations can generally be done using simpler techniques stemming from the regularization implicit in the diffusion operators, but quasilinear waves must be handled with more care given the lack of smoothing properties of the leading order operator.…”