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We present an analysis based on word combinatorics of splitting integrators for Ito or Stratonovich systems of stochastic differential equations. In particular we present a technique to write down systematically the expansion of the local…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…
We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions,…
This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic…
This article is based on earlier papers where an approach based on Taylor expansion and the structure of its leading term as an element of a free Lie algebra was described for the setup of a system of order conditions for operator splitting…
In this article, the existence and uniqueness about the solution for a class of stochastic fractional-order differential equation systems are investigated, where the fractional derivative is described in Caputo sense. The fractional…
This article is devoted to the construction of new numerical methods for the semiclassical Schr\"odinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter.…
Approximate solutions of the Fisher equation obtained by different splitting methods are investigated. The error of this nonlinear problem is analyzed. The order of different splitting methods coupled with numerical methods of different…
High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split…
Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large…
Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric…
Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part.…
For diffusion-reaction equations employing a splitting procedure is attractive as it reduces the computational demand and facilitates a parallel implementation. Moreover, it opens up the possibility to construct second-order integrators…
This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation…
We present a technique, based on so-called word series, to write down in a systematic way expansions of the strong and weak local errors of splitting algorithms for the integration of Stratonovich stochastic differential equations. Those…
We consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative…
We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure…
An error analysis of a splitting method applied to the Zakharov system is given. The numerical method is a Lie-Trotter splitting in time that is combined with a Fourier collocation in space to a fully discrete method. First-order…
We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order…