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

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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 higher-order 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: $u'(t) = Q(u(t)) u(t), t \in (0,T), u(0)$ given. A fundamental requirement for the stability and error analysis is that the domains of the defining sectorial operators $Q(v): D = D(Q(v)) \to X$ are independent of $v \in V \subset X$. The 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.