Convergence of an Operator Splitting Scheme for Abstract Stochastic Evolution Equations

Abstract

In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the approximation for both the original and spatially discretized problems. It is known that the strong convergence of this scheme is classically of half-order, at best. We demonstrate that this is in fact the optimal order of convergence in the proposed setting, with the actual order being dependent upon the regularity of noise collected from applications.

Second-Order Semi-Discretized Schemes for Solving Stochastic Quenching Models on Arbitrary Spatial Grids

“…is problem can be viewed as a simplification of the internal combustion when dual ignition sparks are utilized [2,3]. e partial differential equation problem is first split to one-dimensional subproblems via exponential splitting and then Strategies 1 and 2 are applied, respectively [8,14,18].…”

“…Such a value T σ is called the quenching time which is a possible combustion-reaction time [7,8]. It has been shown that a necessary condition for quenching to occur is max 0≤x≤1 u(x, t) ⟶ 1 − , as t ⟶ T − σ .…”

“…Most of the existing algorithms are constructed either based on the reduced problems, that is, the stationary problems by removing the variable t, or by using fixed mesh steps (cf. [4,7,8,[12][13][14] and references therein). In these procedures, critical values, quenching times, solution u, and the rate-ofchange function u t are often approximated incorrectly.…”

“…Computational procedures are less efficient and less reliable, in particular when multidimensional modeling equations are considered. Although adaptive strategies have been particularly in favor due to their great flexibility and geometric accuracy in capturing quenching singularities involved, orders of accuracies of the spatial discretization of existing adaptive methods have been limited to one due to the unpredictability of the nonuniform grids generated by adaptations [6,8,12].…”

Intrinsic properties of strongly continuous fractional semigroups in normed vector spaces

“…Researchers in numerical analysis have recently benefited from a surge of activities which overlaps with pure mathematical analysis (see, for instance, [3,6,15,16,24,26,28] and publications cited therein). Inspired by this fact and the recent works of Littlejohn and Wellman concerning the investigation of self-adjoint operators in extended Hilbert spaces (cf., e.g., [19,20,21]), in this article we consider so-called strongly continuous fractional semigroups in arbitrary normed vector spaces.…”