Related papers: Word combinatorics for stochastic differential equ…
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 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…
This paper discusses an efficient implementation of the generation of order conditions for the construction of exponential integrators like exponential splitting and Magnus-type methods in the computer algebra system Maple. At the core of…
As an application of the BCH-formula, order conditions for splitting schemes are derived. The same conditions can be obtained by using non-commutative power series techniques and inspecting the coefficients of Lyndon-Shirshov words.
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…
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.…
The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Ito stochastic processes in a neighborhood of a fixed moment of time is considered. The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions are…
We present a class of exponential integrators to compute solutions of the stochastic Schr\"odinger equation arising from the modeling of open quantum systems. In order to be able to implement the methods within the same framework as the…
The article is devoted to the construction of explicit one-step numerical methods with the strong orders of convergence 2.0, 2,5, and 3.0 for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the…
We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications…
We present a practical algorithm based on symplectic splitting methods to integrate numerically in time the Schr\"odinger equation. When discretized in space, the Schr\"odinger equation can be recast as a classical Hamiltonian system…
The article is devoted to the construction of explicit one-step strong numerical methods with the orders 2.0 and 2.5 of convergence for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the…
We study an explicit exponential scheme for the time discretisation of stochastic Schr\"odinger equations driven by additive or multiplicative Ito noise. The numerical scheme is shown to converge with strong order $1$ if the noise is…
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…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated…
The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schr\"{o}dinger equations. However, when applied to certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithm loses…
The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a…
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…
We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves…