document: Stability of W-methods with applications to operator splitting and to geometric theory We analyze the stability properties of W-methods applied to the parabolic initial value problem u ' + Au = Bu . We work in an abstract Banach space setting , assuming that A is the generator of an analytic semigroup and that B is relatively bounded with respect to A . Since W-methods treat the term with A implicitly , whereas the term involving B is discretized in an explicit way , they can be regarded as splitting methods . As an application of our stability results , convergence for nonsmooth initial data is shown . Moreover , the layout of a geometric theory for discretizations of semilinear parabolic problems u ' + Au = f -LRB- u -RRB- is presented
doc_bio_tags: O O O O O O B I O O B I O O O O O O O O O O B I I I O O O O O O O O O O O B I I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I O O O O O O O O O O O O O O O O O
extractive_keyphrases: operator splitting geometric theory parabolic initial value problem abstract banach space analytic semigroup nonsmooth initial data
abstractive_keyphrases: w-methods stability linearly implicit runge-kutta methods